Put simply: It's one of the few ways most men, that are not winning in life, feel like winning in life. It's the closest they'll get to that feeling of accomplishing something. So when "your" team wins, you "win".
PS: Let me start by saying I'm a US citizen and grew up in the US (NYC to be more specific) but I am Spanish in blood and the whole American Football vs Football aka soccer argument is moronic. It comes down to ego. Americans have an ego to protect, it's the "American way". Football is and will always be the No. 1 sport. It brings 10X more money, and 100X the audience. To give you some numbers that can help you understand the compassion between apples and grapes: The Superbowl has about 114 million viewers, and the World Cup has about 1 billion + viewers. Whoever discusses the importance and impact between the two sports is delusional. American Football has a national impact, Football has a global impact. Period. Not to mention Football is older, and its name is derived from actually playing with your feet + a ball not your hands + egg shape thing.
Don't get me wrong, I enjoy watching American Football but there is simply no comparison. Football is a religion.