I thought the same a long time ago, but to be honest I think I was looking at it wrong. I think the reason sports tend to get such support from most people is because it speaks to our primal instinct to compete with one another, which is an instinct that is present in all life from childhood (where you see two kids fighting each other with sticks) to adulthood; just look at males of any species when it comes to getting mates or territory - they'll duke it out with everything they can as the rewards can mean everything.
In sport, we put a lot of emphasis on winning the trophies, cups, titles and breaking records, etc. because it gives us a goal to strive towards and a motivation to become better - both personally and more than anyone before. It's the competitive spirit that inspires us to be all we can be and to go beyond our limits, whether that's on our own or as a team - especially as it gives the people who live in the same county/country a 'hero/heroes' to root for.
On the other side, it's entertaining for many to watch - regardless of whether you're watching two people give it their all in a ring, drivers pushing loud and ridiculously machinery that has no earthly reason to exist to its limits or footballers working as a team to outmatch their opponents; it's cathartic and a nice release that provides a break from real life with all its stresses and hang-ups.
While there will always be people who don't like it (many of us included), you can't deny the impact of exercising our primal instinct to compete in a way that millions (if not billions) can enjoy the world over.