The "-conscious" suffix means that the person is very aware and concerned and knowledgable about the word before the hyphen.
A person who is clothes-conscious is very particular about how he or she dresses, and is probably aware of the latest fashions and knows the names of the major fashion designers. A clothes-conscious person will very strictly follow the unwritten rules of fashion (such as never wearing white shoes after Labor Day). Likewise, a sports-conscious person will follow a particular (or several) sports very carefully, watching games on TV and reading about their heroes in Sports Illustrated. (One personal anecdote I can give of a "sports-conscious" person is my best friend's brother; many times when we visited him he had a radio on in the kitchen that was broadcasting a hockey game. He'd dash into the kitchen to hear what was going on during commercial breaks of the TV basketball game he was watching in the living room. )
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