Same thing here.
For reference my internet is a smooth 14ms to google's servers, and my pc's pretty good, so not problem on that part.
Also it happened twice to me, over 5 days, so maybe 2 games in 30 or something like that.
As for what happens, here's the rundown :
So I select my role, launch the queue, find a game, accept, get in the lobby, pick a champion, wait for everyone to pick a champion, then when the gamer proper is meant to launch ... nothing. The lobby goes to the "try to reconnect" screen and impossible to load in, clicking the button just does nothing.
Then of course the game is remade and at that moment the lobby goes to post-game lobby, for me to enjoy that warning about being afk and my lp losses. Thanks rito.
You can launch a game just after that if you're not tilted by riot's incompetence and cross your fingers not to get that 1% chance of having the game not launching.
Great job riot.