David just bolted and Shane was frozen, so I think Ilya was offering primarily to go after David. Of course that would involve talking to him, and probably absorbing whatever his first reaction was, and Ilya was ready to do that for Shane. But more than anything, he didn't want Shane to be left behind with his panic if David just went away, so he offered to go after him to handle the situation there and then, if Shane wasn't able to do it himself in that moment.
If we wanna layer this, and we do, I think seeing a panicked Holander bolt hit something too familiar, and Ilya wasn't gonna let a bad situation get worse just because two people didn't handle it head on, again.
His father recently passed away, and they didn't have a good relationship, but when Ilya says "I wish he could know me", it's clear he still loved his father and wished he could have had his support, that he could be seen fully as he is by his father, and still be accepted. Ilya never had a chance for that, because of who his father is, but now he's dead, even that wishful thinking is gone. Shane, on the other hand, has support from his parents and here is the chance for him to be seen by them completely and be embraced. And so Ilya immediately jumps to offer help, whatever he can offer, to make sure Shane gets what he never did.
I think it's a bit of what pushes him to meddle in the business of the other queer players too, especially troy, encouraging them to be honest and pursue the relationships that can make them happy and don't waste time, like him and Shane did. To be out and proud and happy in a way he can't achieve yet.
Ilya knows pain and will always do his best to keep those he loves safe from it. And he's taking an active stance, he's not gonna offer to walk away, he's staying (that's not even a question) and if Shane wanted, he'd take the lead of the situation, until Shane got to a place where he could handle it better.
He offers to go to the Hollanders' cottage with Shane to support him, help him, back him up. Ilya and Shane don't need to do any of this alone again, they have each other now.