I can't help you with what exactly went wrong because it could be a large number of things. However, the game is telling you to check the logs in the game directory.
C: Riot Games/League of Legends/logs/
After you get to the logs folder, there are a few different folders that the log you're looking for could be under. My best guess would be the Patcher Logs folder and just look through the most recent one. Honestly, you as a player probably won't be able to figure out what the issue is because it shows issues with the game that A) May or may not even be relevant or B) You wouldn't know what they are anyways due to being displayed in a not-so-user-friendly way.
As a first resort, I'd check this: http://boards.na.leagueoflegends.com/en/c/help-support/mXVGurWN-new-having-issues-patching-520-click-here
If that gives nothing, your best bet would be to submit a support ticket with log files attached.
p.s. Just for future reference, this thread is in the wrong forum. It should be in Help & Support, not Miscellaneous.