I think MOBA popularity comes down to a couple factors:
It's an RTS, but you only need to control one character. Lower skill cap overall.
Most of the action is PvE - creeps, towers, dropping wards without fighting, etc. with maybe 5-15 minutes of direct PvP combat in a 30-60 minute game.
Individual heroes, unit upgrades and leveling/class roles give it an RPG feel.
It has a social element, play with your friends, etc. and you can blame someone else when your team fails. Also F2P lol.
Even back in the Warcraft III days when Dota was in its infancy, I preferred arena combat games like "Advanced Hero Arena X" that didn't have NPC creeps running around, towers, laning, and such. Basically a pure teamwork PvP experience. Well apparently other people didn't think so. That suggests to me that people prefer MOBAs over RTSes or arena combat games because of the lower skill cap and heavy PvE elements.