Mindwipe is a 24th century technology that can erase a part of a person's memories. The process is said to be very painful.
When Michelle Visard abducted Agent 5 from the past, she held him hostage in a missile silo. Since he had gathered incriminating evidence of her tampering with history, she prepared to mindwipe him and then send him back to his era. With the help of Arthur, her plan was averted.
Nonetheless, he didn't avoid the mindwipe: his present self erased his memory before sending him back to the past, as the knowledge of what happened (or would happen) to him could alter history. The events occurred to Agent 5 as they should.
Afterwards, Agent 5 relied on the footage of his JumpSuit in order to "remember" what happened to Agent 3 and to Arthur.