This is just my take on it rather than a canon answer, but I think at first he started wearing them because he was doing anything in his power to appeal to Jack (who had 1) caved in and given him the job he’d been hounding him for the one time Ianto was wearing a suit, immediately after which 2) he explicitly told Ianto “I like your suit, by the way”; I imagine Ianto was pulling every single trick he could think of to get Jack to like him/be happy to have him around, and if making himself look prettier might help do the trick, then so be it).
I do think it started as just that, but the fact that he keeps wearing them after Cyberwoman? I think the reason is twofold. On the one hand, I think he realised that people treated him differently when he was wearing suits, looking and acting professional (and middle-class). I think the suits were useful to him, and even maybe became psychological amour in that they allowed him to present to the world this version of him, a version that was smart (but not threateningly so), polite and respectable; useful to the team and to Jack especially (something he’d want both before and after Cyberwoman: before because he needed access to the Hub and after because he had nothing but his job left at that point). We see him use this in Meat, when he enters the warehouse and runs into a goon (? someone who’d either attack him or call for someone to apprehend/attack him anyway), and he smiles agreeably and reaches into his suit jacket saying “Hello! I have a thingie here somewhere…” as if looking for identification, and the goon lets him because Ianto doesn’t look like a threat, Ianto was acting like he had a perfectly respectable reason to be there, Ianto pre-emptively arms himself to look unthreatening and respectable and exactly like the kind of nice young (again, middle-class) man who is above suspicion unless given reason to the contrary. So the goon responds to Ianto’s eye contact and his friendly (not at all panicked!) tone and he listens to what Ianto’s saying and he doesn’t see Ianto’s stun gun coming at all.
Now, I think looking ‘respectable’ and ‘trustworthy’ is not only useful to him, but also psychologically satisfying, and I think this because of what little we know about Ianto’s bad relationship with his father and the way he lied about him being a master tailor, as revealed in CoE. (There’s more information in some audios, I think, but I’m not familiar with it; if someone is, please share it with us!) By presenting and performing this version of himself (not a lie, please note I think this is a genuine version of himself, not the whole truth but not an untruth), I think he is also psychologically distancing himself from his (lower-class) father, his family and his past. I think his childhood/teenage years were not great and that his relationship with his family then was unhealthy (at least his father, and to some extent his sister, who is shown defending their father when Ianto reminisces about how the man broke his leg when he was a child), and I think that he’s remade himself as an adult, in a way.
He rose from the pain of an unhappy or at least troubled childhood where he felt unprotected and probably helpless because he couldn’t change his circumstances, went through a teenage rebellion where he tried to take control of his life before he truly could become independent enough to change too much of consequence (we know he was arrested for shoplifting and we know he had a goth phase, which let me tell you from personal experience, is often indicative of feelings of angst and dread in terms of the state of the world or one’s own life, a feeling of being different and misunderstood, of society’s norms being off, social mores askew; all feelings that often you can’t fully articulate or even fully comprehend during adolescence), until he was finally old enough to move out, move away and find out who he was, who he could be, away from those negative (at least subjectively to him) influences. He got to make a life for himself, carved out whatever happiness he could from it, and I think that in order to do so, he put as much physical and psychological distance as he could between his new adult life and his old unhappy childhood/family life. As an adult, he finally got agency, the right and responsibility to make his choices, and it makes sense that he’d both want to distance himself from that old life and that he might feel he needed to in order to truly spread his wings. And I think the suits are, figuratively, a way to do that.
On the other hand, and partly due to the reasons I put forth in the previous point, I think he simply grows to like them. That’s it, pretty straightforward. I explained why I think they’re a useful tool to him; and why I think symbolically, they represent his agency and his triumph over the unhappy childhood which he managed to rise above, and over his borderline abusive father and the sister who sided with him. Another, less convoluted reason for Ianto to grow to like the suits could be that Jack likes them. Positive reinforcement can do wonders for making you like something, and I imagine that Jack is the type of lover to make his appreciation known. Even more simple might be the fact that… suits are an aesthetically pleasing fashion choice. They look good. They make the wearer look good; they’re culturally coded as more sophisticated than casual wear and they make the wearer look more sophisticated too. After wearing them for a while it makes sense that he might develop an eye for them and might simply enjoy wearing them, buying new ones, etc. (We see him experiment with colour in the second season, his outfits growing noticeably more eye-catching, so I think this is likely.)
As for what Jack meant in Broken, when he told Ianto “don’t think I don’t know why you wear suits, you don’t need a suit to make coffee”, remember that it is in direct response to Ianto’s accusation that everything they see and do as Torchwood just slides off Jack because he always seems so happy and carefree. Jack responds to that “we all have our masks” or something to that effect, something that implies "we all have our coping mechanisms, and this is mine; I know that your suits and your persona at work is yours”. I always assumed that by that he was referring to what I explained above about Ianto’s performance of this version of himself (of which the suits are the perfect symbol): presenting himself this way allows him certain benefits, it is what allowed him to stay under the radar in the Hub for so long, and it is what allows him to move mostly unnoticed (unsuspected, unexamined, unseen) even after Cyberwoman. The suits (his work persona) are armour, they protect him from being too vulnerable. But they’re also an obstacle, they keep others at a distance, and he remains unknown, apart, alone. He begins to breach this distance at the end of Broken with Jack, and I think we start seeing him open up a little with the others too after it, though more slowly.
I’ve mentioned this before, but I think that Ianto didn’t fully trust the team - and didn’t fully become part of the team subjectively (i.e. in his own eyes; he didn’t feel like part of the team) - until season 2, after they were forced to work together to make up for the void left behind with Jack’s absence. In season 2, starting from the very first episodes, we see Ianto be more open around the team, we see him volunteer sarcastic comments more often, and we see him be friends with the others in a way that wasn’t there in season 1. This is partly due to Ianto’s own trauma, remember he was still processing the terrible loss of Lisa (and the role that Torchwood and Jack played in it) all during Broken, and he also mentions there that he doesn’t feel like part of the team. He isn’t going out on the field up to Broken and the rest of the team apparently make little to no effort to include him or to even to get to know him. And Ianto is ill-equipped to know how to resolve that, even as he confesses to Jack that he does want to resolve it, that he wants to be part of the team. And pretty soon after that Owen starts going off the rails (first in Combat, and then in a way that affects Ianto and the team much more directly in Captain Jack Harkness and End of Days), which puts him and Ianto at odds, so broadly it makes sense for Ianto to still feel like an outsider to a point.
But it is in Broken that we see Ianto make the resolution to try, when he rescues Jack despite his first thought being having his revenge for Lisa by letting him be taken, he’s choosing Torchwood and his future there over clinging on to a past he can’t go back to; but even before that, when he opens up enough to ask Jack to take him out on the field, when he admits that the others still know nothing about him and that it bothers him that they don’t seem to want to learn, when he’s touched when Jack calls him his friend… we see the journey Ianto goes through and we see the exact moment he chooses to let go and to move forward with Torchwood. Even if there’s a setback later with Bilis’ manipulations, recovery isn’t always a straight line, and that setback doesn’t negate the incredible strenght he displays when he chooses to move forward, to make an effort to get what he wants, even if it means making himself vulnerable and letting go of the pain (and guilt) of not being able to save Lisa. And letting go of the pain also meant letting go of the fantasy of that life which Lisa represented, letting go of any last vestige of hope that he could have a life like that. (I’m admittedly reaching here, but I think Lisa represented a more traditional sort of victory for him: the kind of lover that society would acknowledge as representative of a normal life vs Jack the immortal man as representative of the kind of life Torchwood offers: different, secretive by nature and by necessity, always in the fringes, full of joy and wonder and excitement and sorrow which you can’t share with anyone who isn’t in the know but worth it anyway, wonderful anyway, and how could you go back to normal once you’ve experienced it?)
Anyway, these are my thoughts on it. As you can see, I have plenty :D I’d love to read other takes, whether they agree or disagree with me, so if you feel like sharing do tag me so I see it!