Why are some champions considered to be better at blue/red side?

Haidros answer is correct but pick and ban popularity aren't the only reason. There are still 2 (or 3 depending on how you see it) things in the game that show benefits or disadvantages to a certain team.

Dragon
Dragon is closer to the Blue team but leaves more space to fight when the purple team approaches from their jungle. This is especially important during early and midgame since everyone wants the winged lizards head during these game-stages. Also champions like Lee sin that can just jump into the pit to steal it are slightly stronger when played on the blue side, since the jungle behind dragon is Blue-Territory and thus much safer for Blue team. Trying to steal a Dragon during Early and midgame is usually much harder for Purple Team than for the blue team. These things don't apply to baron, since you usually kill him in lategame and there the jungle territory isn't that impactful anymore.

The Tri-Bushes
This is by far the biggest difference when it comes to team-specific champion viability. There are two Tri-Bushes, one on the bottom lane and one of the top lane. Both team have a disadvantage there. The toplane for Purple side and the Bottom lane for Blue side. The tribush allows you to cut off or siege a laner which makes escaping ganks extremely difficult. You usually want to pick a champion with a good gapcloser or disengage on the respective handicapped lane. At the same time champions that have hard CC or even heavy Knockback spells (Lee-Sin, Janna) are extremely strong in ganking said disadvantaged lanes.

For your point about banning Alistar:
If you bait a fight as alistar and you manage to headbutt your opponent further into your lane, there's almost no chance of escaping an incoming jungle (+ midlaner) gank from the tri-bush. That's the main reason why he's so scary.

You're right, most of the time the side you are playing on doesn't really influence the strength of any champion. (Debatable with Nautillus maybe if you consider walls)

The reason why the casters said that red (purple) side should ban alistar is so that blue side don't get it.

In draft mode (bans included), blue side pick the first champion after each side bans 3 champions. Purple then pick 2, blue pick another 2, purple pick another 2, then blue pick their last 2 champions, while purple pick their last champion.

Thus blue side is most likely to pick the champion they most desire first to keep it away from purple side. And Alistar is great top lane and is well wanted from both teams.