An ejecting mechanism in the game, where the cockpit would launch out from the back of the torso of the mech when the mech gets destroyed.
In the arena, this could show for either of the mechs, because the arena is supposed to be a place for Pilots to fight and become strong enough to fight Campaign bosses.
And in campaign, this animation could show only when your own mech gets destroyed, because when it does, you can respawn or retry the mission again and be able to do it again.
Also, the mechs you fight in Campaign are evil, so it wouldn't make sense if they were able to escape in that way.