In every standard Objective Mode level, there will be special events that occur at random while the Objective Mode timer is running.
One of these is "A mouse is loose. Catch it". This requires you to find a mouse that is roaming around somewhere in the level (you don't get a waypoint to help you find it) and catch it.
You "catch" a mouse by swatting or pawing it. If successful, you'll get a yellow (upgrade) or purple (power up) box as a reward and you'll also unlock this trophy.