I need help understanding how to explain why GPE is equal to work when friction is ignored. For example in the question:

In order to look for wind, a grinder hoists the bowman (mass=75kg) to the top of the mast (height = 34m). Ignoring the effect of friction, explain why the bowman’s gain in GPE is equal to the amount of work done by the grinder in getting him to the top of the mast.

GPE is one form of energy. Work done is another name for energy. As he is hoisting the bowman vertically, he must be applying a force, in a direction (upwards). Knowing the distance allows us to calculate work done (Energy):

Work done = Force x Distance

If there is no friction, there is no energy being transformed (lost) into other types, such as heat, or sound energy. Since no energy is being lost, then whatever energy the grinder spends, must be the GPE gained by the bowman.

We can also look at this another way. Looking at the equations we have:
GPE=mgh and W=Fd

We also know that weight force (F) can be calculated by F=mg
Since we also have mg in GPE=mgh, we can replace mg with F
This gives us:
GPE=Fh
h is just a distance vertically, so we can replace it with d:
GPE=Fd
Remember W=Fd, so we can now hopefully see that (when dealing with vertical distances) the change in GPE and Work done are the same.

Hope that helps, fire away with more questions if you’re still not sure. This video guide might help with the concepts you are asking about.