Why are characters banned in tournament play for DotA-likes?
Why are characters banned from DotA-likes in tournament play during selection? Are there characters that are actually imbalanced; why not just balance them?
If it's just for the sake of another level of meta-game strategy, then where did it come from?
No, this isn't related to balance, and is actually strategy related.
The origins of banning heroes comes from a match type called -CM, or Captain's Mode in DoTA. In Captain's mode, teams take turns banning heroes they don't want the other team to use whilst picking their own heroes for the match. One person from each team (the person in the first slot) is the one who makes these decisions, so you usually only play CM with a group of people you're familiar with since it requires a lot of coordination.
In professional level play, this becomes extremely important as you want to prevent the other team from picking heroes that they are adept at, or countering the heroes you picked.
For example, say you wanted to go with an "all push" strategy that relies on pushing down lanes to end the game fast (with the drawback that if you don't end the game early, the other team will easily overpower you over time with their "late game" heroes). Your team will probably want to ban any heroes that can counter these pushes, or easily kill their heroes.
While balance/imbalance can certainly have a factor from patch to patch, professional level play sticks to a version of the game that is considered "stable" by the community.