another book by stasys jukna who was cited in the other answer has some key "applications". as always applications can be subjective, but in TCS a key application many would agree on is complexity class separations. there are about 30 references to ramsey graphs in this book as counted by a pdf search.
jukna makes the case that ramsey theory is about large structures guaranteed to contain some kind of "feature" and these can be shown to exist by shannon-style or shannon-reminiscent counting arguments, but building explicit constructions is very difficult and rarely accomplished in the literature. this is apparently highly analogous, possibly even directly linked, to the inability to explicitly construct complex circuits even though they are known to exist. construction of such circuits is key in complexity class separations. this is further emphasized by constructions in juknas book that directly tie graph complexity to complexity class separations (although so far not directly through ramsey graphs).
therefore this (same?/crosscutting?) phenomenon seems to be something like the "dark matter" of computer science. it is known to be there, it can be indirectly measured, but cannot be directly exhibited so to speak, and it is yet mysterious. jukna implies ramsey type thms or constructions could possibly be a bridge-type thm for complexity class separations (see sec 1.7).
see also sec 1.5, "where are all the complex functions"? however it is not so straightfwd, in constrast see also sec 11.7 where he uses ramsey graphs to show how "combinatorially complicated" graphs eg ramsey graphs are not nec "computationally complicated".
 Boolean function complexity, Advances and Frontiers by Stasys Jukna, 2011