Keisler measures are finitely additive measures on the Boolean algebra of the definable subsets of some model.
They were introduced by Keisler in 1987 and did not receive much attention until Hrushovski, Peterzil, and Pillay used them to solve an important conjecture.
Since then, Keiseler measure have become a mainstream topic in model theory.
Is there a more accessible result than the Hrushovski, Peterzil, Pillay theorem cited above that shows an application of Keisler measures?
One very nice application is the distal regularity lemma, which is a "Szemerédi regularity"-type statement for graphs definable in distal structures (the regularity is much more regular than what the Szemerédi Regularity Lemma gives us for arbitrary graphs).
The distal regularity lemma is due to Chernikov and Starchenko: Regularity lemma for distal structures, generalizing previous work for semialgebraic graphs (i.e. graphs definable in the real field, a canonical example of a distal structure).
Shortly thereafter, Simon give a much quicker proof of the main result: A Note on "Regularity lemma for distal structures". Both proofs use Keisler measures in a crucial way. You might find Simon's note more "accessible" - or possibly too terse! Either way, the Chernikov-Starchenko paper contains a lot more background and motivation, so it's worth a read.