Well watching football or any sport for that matter is something which can hardly be put into words. You develop a passion for it, is fascinated by all its aspects, like to analyse and study (talking about the select properly passionate people here).
You feel a sense of belonging to the sport, to the club, exactly like may people do for art or music or anything similar. Football for one is the prime example of how the whole set of human emotion is summed up in 90 minutes. You feel extreme joy when your team wins, sadness and a sense of loyalty when it loses.
We feel represented, your ideology matches the club's is one reason. In simple words every sport has its own attraction, so does the football.