I've heard rumors about it loading all the game files into the RAM, but that seems a bit off, given that League is about 10 gigabytes on my HDD and I only have 2 gigabytes of RAM. I'm pretty sure network latency isn't the cause either; when I play at a friend's house that gets ~10 ping, my loading time is the same when I play at my house with ~70 ping. What exactly is going on?
Rito answer pliseru, it'd be great to hear from a dev.
I've heard rumors, that the game is "Loading" during the loading screen. Not confirmed :( 
Loading time and ping are in no way related.
During the loading screen the client is loading all of the assets for the map, champion information and all other relevant information required to play the game and places this into what is often called a "runtime environment".
The speed at which this happens is based purely on the performance of your computer. Mainly CPU/RAM speeds and hard drive read speed. Ping is just a measure of the amount of time it takes your computer to contact the riot servers and receive a response.
Basically all the assets and files that are required to run the program are copied from the hard drive and placed into your computers RAM or running memory.
I should also add that it is not loading ALL of the files belonging to League of Legends, just those that are required for the current game. Relevant champions, relevant map, neutral monsters, etc. etc. For example, when you play a standard 5v5 game on Summoner's Rift, it doesn't also load Twisted Treeline or any related information to it.
10 gigabytes is the size of 100+ champions but you only load 10 champions so you just load 1 gb.
The game is actually ~3.5GB (the rest is patches for whatever reason, because we all need patch 4.1 still amirite), so when it's loading 1/4 maps and 10/120 champs, a game very easily fits inside 1GB RAM.
And it's definitely loading from disk into memory. My friend just replaced his hard disk with an SSD and his load time improved 3x. Additionally, my Mac (Macs have cache memory, so when the game ends most of the files still stay in the RAM) will load a game roughly twice as fast on the second and all subsequent games.
