Broadway can act so oafish sometimes, that it can be easy to forget that he actually is smart to. He's smart in the way of emotions imo, and I think that lends itself to this moment. I don't think he knows how to put it into words here, but I think he's picking up on the way Elisa is being a bit rash herself. Maybe not treating it like a game but also not giving it the seriousness it deserves.
"I'm alone on this one," she says to him to tell him to back off, but she's also telling on herself. As Katrani pointed out in another comment, Elisa never wanted a partner. I think that's very much what we're seeing here. Elisa didn't want a partner because she should be able to do things on her own. It's in a similar vein I think in why she hoards her knowledge of the gargoyles to herself past what is reasonably necessary. She already has this bond with Matt, but she doesn't tell him about them; her brother was in serious danger and repeatedly refused to tell him about them. She hasn't shared their presence with anyone in her life important to her because she wants to be the lone human. She wants to carry the weight of it all and feel accomplished in that she can handle all that, that she is special. She is such a good detective that she can rescue Matt.
But even her captain was worried about this job and asked Elisa call for backup, but Elisa won't do that. And I think, in his own way, this is Broadway calling her out for that. And if she really wants to save Matt, isn't it worth putting in the energy to letting Broadway help, even if that means taking extra time to tell him how its done and what she needs of him instead of him guessing?? He is a powerful tool that she would be absolutely remiss to wave away when so much is on the line.
And I think, if he'd called her out as directly as I'm stating it here, she would have dismissed him and said it was ridiculous. But the way he did put it? It's hard for her to ignore the point he's made, so she caves. (At least I'm fairly sure this is her caving)