Having bans in Bo3 sets basically creates an artificially shorter set that functions like a Bo5. By eliminating the matches that would take place on both player's best stages, the Bo3 essentially assumes that both players will win on their strongest counterpick, and they only play out the matches on the more even stages.
For instance:
Fox v. Marth
Fox will most likely win on DL64, and Marth will most likely win on FD.
Bo5 (no bans)
Game 1: Strikes to BF. Fox wins.
Game 2: Marth counterpicks FD. Marth wins convincingly.
Game 3: Fox counterpicks DL64. Fox wins convincingly.
Game 4: Marth counterpicks FoD. Marth wins, but it's a close match.
Game 5: Fox counterpicks PS. Fox wins, but it's a close match.
Bo3 (with bans)
Game 1: Strikes to BF. Fox wins.
Game 2: FD is banned, Marth counterpicks FoD. Marth wins, but it's a close match.
Game 3: DL64 is banned, Fox counterpicks PS. Fox wins, but it's a close match.
Do you see how it acts nearly identically to the first set? But instead of two convincing wins on strong counterpicks (which is fine; I'm not against strong counterpicks in a longer set) we instead cut out those matches, assume each player won on their respective best stage, and then basically the two wins negate each other. Only the closer, more balanced matches on more winnable counterpicks are played. The set is shortened but the integrity of the outcome remains relatively intact. Either Fox or Marth could have won on either player's second-best stage, so the first game is not all-powerful in determining the winner of a Bo3 set. Compare to no bans:
Bo3 (no bans)
Game 1: Strikes to BF. Fox wins.
Game 2: Marth counterpicks FD. Marth wins convincingly.
Game 3: Fox counterpicks DL64. Fox wins convincingly.
The set is much less interesting, and the winner is essentially determined after Game 1. I know that for this to be precisely true, we have to assume that the Marth isn't somehow godly on DL64, and likewise the Fox isn't godly on FD, but for a set that must be shortened due to time constraints, Bo3 with bans is the best way to simulate a true Bo5.