How to cite earlier work, discovered only at the end, that claims similar results to a recently completed project? At the end of writing up our manuscript, we discover that there has been another earlier study in a very different context (but using a physical model very similar to ours) that claims a similar result. 

We were not inspired by this earlier work and we could not have found it until we had our results; so this is not a case of not having done proper literature survey a priori. 

So, do we cite them within the main text's introduction and results despite not having been inspired by it at all? Or do we cite them during the concluding remarks highlighting the similarities?

In either case, our results certainly complement theirs.
In mathematics (maybe in other fields too), one would put at the end of the introduction a statement like "After obtaining the results in this paper, we learned of related work by X.  In particular, X obtained ...." Here the "..." would be a description like "a stronger form of our Theorem 7" or "a weaker form of our Theorem 7" or "a result related to our Theorem 7" or whatever it was that X actually got.

In computer science (maybe in other fields too), papers often have a separate section called "Related Work", and information of this sort would naturally go into that section.