I am whole but incomplete
This means a skeleton is whole considering it as a structure. But it is just a part of the body. So it's incomplete.
I have no eyes, yet I see.
You can see, and see right through me.
This means skeletons don't have eyes - just the holes. By yet I can 'see' it refers to the 'see' in the word 's'k'e'l'e'ton. The same is meant by 'you can see', and 'see right through me'. Which makes it clear that we can see the word 'see' in (right through) 'skeleton'.
(Or as mentioned in the comments: people perceive a skeleton to "see" through its eye sockets, even though it no longer has eyes. And people can see through the gaps in a skeleton.)
My largest part is one fourth of what I once was.
The largest part of a skeleton is the femur, which is one fourth of the body (which the skeleton once was) in length.
Hope it's clear now.