"Stakes" is a noun meaning "A sum of money or something else of value gambled on the outcome of a risky game or venture." (Definition borrowed from Oxford Dictionary.)
"Style stakes" refers to the "gambles that celebrities take at high profile events by daring to where something that will attract attention or set a trend in fashion."....
Style stakes = the clothes that famous people where at high profile events, because of the undeclared but real competition among celebrities to attract attention by what they wear.