From my memory:
The Institute of War (IoW) is an organisation on Runeterra that was created to stop the use of the destructive and uncontrolled runemagic, for it threatened to destroy the world. It was created when leaders, in particular those of Demacia and Noxus, acknowledged the dangers this magic presented to the world. With the two most powerful city-states supporting it, others swiftly followed suit.
Runemagic was however a pillar of how wars were fought. so the Institute decided to offer an alternative to the classic war to help city-states cope with the sudden 'ban' on runemagic.
They came up with the League of Legends: An arena where each faction sends in powerful champions that will fight battles on battlegrounds, usually located near warring nations for convenience sake. These battles were swiftly adopted and became highly popular, to the point where they were broadcasted to all states for the public to watch and where they were even used to settle more minor disputes with less experienced summoners.
Personally I would've liked it if Riot had just gone back to the institutes roots and made it about stopping uncontrolled runemagic, offering an alternative on the side. Nerf the runemagic summoners can safely use so they can't hold up against a champion, and it would've been a lot more interesting I suspect.
Would've also made the whole collecting champions more sensible; collecting 'messengers' to send out to stop people that defy the Institute.