Why does it need to be expanded if the input has already been compressed/padded?

You state that is expands it into an array of 64 x 4 bytes which turns into the message schedule.

From what I'm seeing, the data in the message scheduler is also part of the data you're compressing but it is somehow separate from the "compressor" as labeled in the diagram.
This "Expander", which is part of the message scheduler, takes in "Wt", a chunk of data from the message schedule and "Kt", a predefined constant. I assume I understand this correctly?