I think in this sentence, 拍手 is in its noun form. And since it’s しすぎ, that would mean “overdo” (instead of “to overdo”). So I would translate it literally as “overdo of clapping” or more succinctly “overclapping” (which is the same as “clapping too much”).
In this sentence の functions as a substitute for が. However since the noun, 拍手, acts as a modifier for the rest of the clause ending in 手, の is used instead. Changing the particle does slightly nuance of the meaning, though.
@siballah で is a particle that can indicate cause. One way to translate this is to say “due to”. I think @mrsaturn’s more literal translation demonstrates this well, which I’ll quote below (emphasis mine):