When the life totals are exchanged, each player gains or loses the amount of life necessary to equal the other player's previous life total. For example, if Player A has 5 life and Player B has 3 life before the exchange, Player A will lose 2 life and Player B will gain 2 life. Replacement effects may modify these gains and losses, and triggered abilities may trigger on them.

If a player can't gain life, that player can't exchange life totals with a player with a higher life total. If a player can't lose life, that player can't exchange life totals with a player with a lower life total. In either of these cases, neither player's life total will change.

We both know how it works, just the idea is that instead of making it life gain and loss, why couldn't you just switch it and not have it register as gain or loss.

Jelly has it right. Setting a life total to a particular value works by having the player gain or lose the appropriate amount of life. This means that Erebos's ability effectively reads "It is impossible for your opponents' life totals to go up."

Since it's impossible for your opponent's life total to go up, the part of the exchange where their life total would go up is also impossible, and if part of an exchange is impossible, all of it's impossible, so nothing happens.