To my knowledge there is no simple proof of the bulk-boundary correspondence. That's why it's so surprising and interesting. It is really weird that there is a correspondence because they are different setups - the edges are a property of the finite chain and topological invariant is a property of the bulk/ infinite chain. At the same time, it is really at the crux of topological phases, and this correspondence has been verified over and over experimentally and numerically, even if we don't have a simple answer of why the correspondence exists. I think there is some heavy maths like K-theory that have tried to explain this correspondence but I don't understand it.
One of my supervisors sent me this paper 'Edge states and the bulk-boundary correspondence in Dirac Hamiltonians' https://arxiv.org/abs/1010.2778 which proves why the topological invariants of the bulk lead to edge states but I haven't looked at in depth myself yet.
On a rather random note, there is a really nice set of notes on this website: https://topologicalphases.wordpress.com/lecture-notes/. At the end of Lecture 6 (on the SSH model!), they show why the winding number is equal to the number of edge states for 1D gapped chiral Hamiltonians. It is password protected though. I got access by cold-emailing them and Tomoki Ozawa was nice enough to give me the password (I am a random student and never met them) so maybe you can email them too if you like. They are really nice and clear notes. They consider a special case that has edge states and use the fact that other Hamiltonians with the same winding number can be continuously transformed into the special case (but this also uses homotopy theory which they don't explain in the notes - so the proof of this seemingly simple correspondence is not so simple).