It's not too hard, it really does just require playing the game. A simplified way to describe it is that you start with 13 tiles, then draw one and figure out how to get to a set of 4 sequences/triplets and one pair, decide which tile is useless to you at that point and discard it. Every player is basically trying to make a legal hand by doing this and usually takes around 10 turns to do it on average.
The calls you can make (Chi/Pon/Kan) can help speed up your hand. You call Chi when you want to steal a tile to make a sequence (you have 2-3, someone drops 1, you call chi and steal it). You can Pon for Triplets (2-2, someone drops a 2) and Kan for when you a triplet and someone drops another (2-2-2, someone drops a 2). These all act as a permanent sequence in your hand you cannot remove. You call Ron if someone discards your winning tile, and Tsumo if you draw it yourself.
That's pretty much the basics without going into legal hands, calculating points and things like Han. You can understand it the more you play. If we ever get some more traffic around here I wouldn't mind hosting some online games via Mahjong Soul or whatever.