ARAM, to me, helps me learn the different champions in League, their abilities, their cooldowns, their power spikes, and their roles in a teamfight. It forces me to play different champions, some I do not necessarily like (For example Yasuo, I got him because I like Japanese Samurai culture but I would just go in and die so I started to hate him but as I kept getting him in ARAM I started to understand his flow. Im nowhere near a god but I am improving and getting kills without giving them up as often and it makes me enjoy playing him). As a support main I would not get the opportunity to play these champions on the Rift. (Support Yasuo coming soon WOOT! ;} )
ARAM allows for more games, as you stated, and it is all teamfights so I believe I get more experience say learning how to hook an enemy through minions with Thresh and learning how to hook past tanks to get the delicious squishy adc.
ARAM also shows me my role, depending on the champion, in a teamfight. Instead of going all in against tanks with Katarina at the start I am learning that assassins have their time and window to go in and until then to stay back. Another example is instead of 1 v 5ing as a Tank I have to have my team regardless if its a good initiation or not. Without follow up its useless. So I believe it improves my teamwork.
In short: I main support but I want to be a diverse player and ARAM is perfect for it. I am still learning but I believe ARAM offers many things such as lessons in knowing your enemy (forcing you to play different champs), knowing your role in teamfights, knowing your champion pool and diversifying that pool.
