In repairing an equipment that has a broken circuit board, you must ground yourself to the body of the equipment your working on to prevent what from damaging??

The components on circuit boards can be sensitive to static electricity. You generate static electricity as you move through the air, when you rub certain materials, and the like. The static electricity can burn out sensitive electronic circuits mounted on the circuit board.

If you wear a "grounding strap" that is connected to the same "ground" as the equipment you are working on, the static electric charge is drained from your body. This prevents damage to the sensitive circuits in the electronic chips.

As a rule, you should ground yourself using a grounding strap connected to "ground" through a 1 megaohm resistor. That way, if you are working on "live" equipment and happen to touch a higher voltage area, the resistor will prevent you from being electrocuted.

I don't recommend, but you can touch metal case of what you are working on to discharge any Static, then you can safely touch component without destruction, any walking, sliding, moving of yourself or board will need an additional touch to ground(case) to discharge any static.
Over 35 years in Electronics, Haven't destroyed anything yet.
Only some electronic parts are suceptible to destruction by Static.

In repairing an equipment that has a broken circuit board,, you must ground your self to the body of the equipment your working on to prevent what from damaging?????

Your homework.