As in "I had that happen before." Why is this correct? Does this mean that something has happened to someone or that you made/let something happen?
I had that happen before
In this sentence, that is referring to the thing or event that happened. It's not specified what caused the event, or who or what it happened to. It just happened.
The person is saying, in effect, I have had previous experience of this event, and the assumption is that the person he or she is speaking to understands the subject under discussion.