I've been playing League of Legends a LOT over the past year or so. Great, easier to pick up version of DotA.
Why is DotA so popular, I never really understood
It combines so many different genres.
You have strategy in how your team sets up, who goes on which lane with which heroes and which items and what their long term goals for the game is. A long term goal in killing a single building by getting to a certain point.
You have RPG in levelling, killing, bosses and items. You can also play many different roles, things like farming carry, ganking carry, ganker, support and pusher.
You have multiplayer, so no game is the same. You can play with the same 10 heroes on the field used by the same players and the game will turn out very differently due to the mind games and the number of strategic options you can use in them. Something as simple as changing who lanes where makes for a completely different game, even if both sides make the same switches.
You have tactics in that you have to judge what to do in each moment, do you try to hurt the opponent, mess up their farming, farm yourself, stay back or gank some other place?
Most of all it is popular due to the map being so good, not having symmetry makes each game different. Having decently balanced heroes and items in huge numbers in a team game means you can pick different combinations to break what the opponent is doing at the pick phase.
The last thing is the most important, the skill roof is so high that it hasn't been touched yet. It is also updated enough that the slightly superior strategies are never perfected and leave the game dead. Each version allows something else to be slightly stronger allowing a dynamic player/team to shine over time. One person can not win a 5v5 alone, two people can if the other three play not to lose. I've even seen 3v5s won where both teams were of decent level (one of course being much higher).