First of all: there is currently no lore in League of Legends. The current narrative team threw all of it out some seven months ago and didn't replace it with anything except a pile of contradictory promotional trailers. I'm not going to go into further detail because there is no point; scrolling through the forum will still expose you to very regular expressions of outrage. Needless to say, neither the current narrative team nor the content they put out are very popular among the lore lovers.
That said, there were actually detailed explanations provided by the original lore team in the Journal of Justice. Here they are:
Summoner/Champion Connection:
Summoning is the most challenging of all magical disciplines. There is far more to it than simply relocating a being from one location to another. A summoner mentally and physically links with their summoned being (champions, as we call them), allowing for a fluid symbiosis. A champion is not a puppet of the summoner; instead, the champion is the executor of the will of the summoner. A champion is in full control of themselves, but they are fully aware of the importance and the necessity of the summoner’s will. A champion’s duty is to execute the will of a summoner with the utmost precision and execution.
Bear this in mind: a champion is never forced to be a part of a battle arena where their loyalties and beliefs are not considered. During matches where city-states are resolving disputes, a champion will never act against their own interests. GarenSquare.png Garen, for example, would never charge out of the brush at a fellow Demacian if the match were about resolving a dispute involving Demacia.
Fields of Justice:
Constructing a Field of Justice is a truly remarkable feat, and the magic at play is beyond the comprehension of all but the most gifted of magicians. Once a battle begins, the Field of Justice becomes a kind of microcosm. Within this sphere of influence the champion and summoner exist as one shared mind.
This does not mean that the champion is not in mortal danger. Death severs the connection between summoner and host champion and is an extremely traumatic experience for both parties. After the summoner has recovered, they can reenter the battle arena by completing the summoning ritual again. At this point the champion is revived on the summoning platform, once again entering the fray at the risk of their own life. For this precise reason, champions and summoners are a rare breed.
Minions:
Minions in the Fields of Justice are animated constructs that, once unleashed, follow pre-planned patterns of behavior. The summoners – either those controlling champions or powering the Field of Justice – have no control over the minions. Rather, the summoners who are assigned to the maintenance of a battle arena focus on quickly and efficiently constructing the minions. Constructing each minion takes a minuscule amount of magic from the nexus it spawns from. These summoners, who are usually apprentice summoners that reside within the Institute of War, are in many ways the engine that allows the battle arenas to function. All summoners who come to the Institute of War will spend time performing such tasks.