Use when someone does more than what is expected or required. She put herself out to bake a cake at midnight for the party.
Pro Tip 2/3
Emphasize Effort
Highlight the amount of effort, especially if it's voluntary and for others' benefit. You've really put yourself out organizing this charity event, thank you.
Pro Tip 3/3
Not Obligatory
Implies the effort made wasn't a requirement but a choice. He put himself out to drive her to the airport, although he didn't have to.