What Went Well
The reason we ask what went well is not so that we can pat each other on the back and inflate our egos. One of the keys to a successful agile project is consistency in the velocity of the team sprint over sprint. One of the best ways to do this is to identify what lead to success so that the team can capitalize on it in the upcoming sprint(s).
What Didn't Go Well
As with most growth being realistic, open and honest about our mistakes and failures leads to not repeating them. The purpose of identifying what didn't go well is NOT to lay blame on any particular person or people. Identifying what didn't go well helps us to avoid or correct them those mistakes.
What Could Be Done Better
At first this may not seem different from identifying what didn't go well. The real difference is that we recognize things that may have gone well but could have gone better. Asking what we could do better is an opportunity to identify gaps in our planning, our process, and our estimates. This question should lead directly to actions that can be taken to increase or maintain the teams current velocity.
Because the sprint review typically takes place before the planning of the next sprint asking these three questions provides the team with the opportunity to make changes that give them a higher likelihood for success before the next sprint starts.