"Fill your boots" means that something is so good and such a great deal that you should take the opportunity to get as much of it as possible. Your pockets, bags and suitcases would not be enough, the deal is SO good that you would also take your boots off and fill them. Not literally of course, but that's where the expression means. It can be used for any particularly good offer, regardless of whether the items are physical and actually would fit in your boots... For example, if British Airways was selling flights to London for £10/return, you'd say "fill your boots" (implying that you would want to buy them, many of them in fact).

I have never heard the expression "fill your boots" specifically, but " having big shoes to fill" is a common expression. "Having big shoes to fill" refers to when someone is replacing another person in some role or capacity who had performed very well in that role. So in your phrase it probably means to fulfill the duties ones position or role.