Ultraproducts are useful in certain applications to combinatorics, with a famous example being Hrushovski's work on finite approximate groups, followed by the structure theorem of Breuillard, Green, and Tao (also using ultraproducts).

The general idea is that instead of proving an asymptotic statement about all finite objects of a certain kind, one instead proves a companion result about a single infinite object. For a toy example, we can prove Ramsey's theorem for finite graphs as follows.

Given an integer $n$, we want to prove the existence of an integer $R(n)$ such that any finite graph with $R(n)$ vertices contains a complete or independent subgraph on $n$ vertices. Suppose not. Then we have a fixed $n$ such that for every integer $k$, there is a graph $\Gamma_k$ with $k$ vertices containing no complete or independent subgraph of size $k$. Let $\Gamma$ be a nonprincipal ultraproduct of the family $(\Gamma_k)_{k\geq 1}$. Now $\Gamma$ is an infinite graph. One then proves a lemma (Ramsey for infinite graphs): any infinite graph contains an infinite complete or independent subgraph. In particular, $\Gamma$ contains an complete or independent subgraph on $n$ vertices. This latter statement is expressible by a first order sentence, which then must hold of some (in fact infinitely many) $\Gamma_k$, a contradiction.

This example is good for illustrating the method in a simple way. But it is perhaps not very satisfying because in this case, the infinite statement (Ramsey for an infinite graph) is not significantly easier to prove than the initial asymptotic statement (Ramsey for finite graphs). In the real life examples showing up in research papers, this is not the case. Often the point is that methods from other fields, such as model theory, can be used to attack the infinite statement in ways that are not clearly adaptable to finite structures. This is certainly the case for the work on approximate groups mentioned above where, in addition to model theory, there are also very nontrivial results about locally compact Lie groups in play.

One downside to the ultraproduct approach is that the statements proved are qualitative by nature, rather than quantitative. For example, the proof above of Ramsey's theorem gives no information about a bound on $R(n)$ in terms of $n$. In some cases, if one understands the "companion proof" for infinite structures deeply enough, then one can see how to finitize the proof and carry it out in a quantitative way. For example, the proof of Ramsey for an infinite graph finitizes in a clear way to a direct proof for a finite graphs yielding a bound for $R(n)$ on the order of $4^n$. In other cases however, the finitization of the infinite proof can be much more challenging. Something I like about this kind of work is that even if one is able to replace the infinite qualitative proof with a finitary quantitative one, it is often the case that the model theoretic contributions of the infinite proof were instrumental in cracking the finitary proof. On the other hand, there are some examples where a quantitative finite proof is still open (e.g., the work on finite approximate groups).