Any time the heroes get to save the day in a big, awesome manner.
Say that the poor Damsel in Distress is looking her Fate Worse Than Death dead in the face, and is resigned to it, because she knows that Nothing Can Save Us Now...and then, boom! The heroes kick down the door and proceed to stomp the bad guys into the ground.
A Ragtag Bunch of Misfits will often have a Misfit Mobilization Moment before becoming the Big Damn Heroes. If they were busy with something else, but saw the crisis and dropped whatever they were doing in order to save the day, it's a case of Dudley Do-Right Stops to Help.
But note that a Big Damn Heroes moment doesn't have to succeed. The villain may declare "You Are Too Late." Or sometimes the heroes' actions end up making things worse.
Often this can involve Gunship Rescue, I Got You Covered, Horseback Heroism, Roaring Rampage of Rescue, Just in Time, or even Changed My Mind, Kid.
Compare The Cavalry (it's the heroes saved in the nick of time), Contrived Coincidence, Bad Guys Do the Dirty Work, Villainous Rescue (when the villains are like this Trope).
If this is done in an anticlimactic way, then it's a Deus Ex Machina, and dooms it bad, of course. When done right, it is a sure-fire way to get a Crowning Moment of Awesome.
If the heroes just finished a training arc and show off new powers, it's Look What I Can Do Now!.
Super-Trope of Roaring Rampage of Rescue. Often leads to a "Hell Yes!" Moment.
Examples of Big Damn Heroes are listed on these subpages: