One purpose of try and mitre squares is to check that angles are correct. Try squares can check both internal and external right angles and mitre squares can check 45° and 135° angles. It is important that all angles are correct or your whole project may be wonky and not fit together properly.
The other purpose of try and mitre squares is to measure and mark angles.
These purposes open themselves to a multitude of tasks. A try and mitre square may come in handy for any DIY task that incorporates right angles.
Examples of uses for try and mitre squares could include: framing, stair-making, shed building and many other carpentry tasks.
Try and mitre squares are also useful for checking the angles on cutting machinery, like bench saws.