Prove that there are at least 100 pairs of usable boots

If any size category $A,B,C$ has equal numbers of left and right boots, there are the required $100$ pairs of usable boots and we are done.

Otherwise every size will have a majority of one foot. Since $101\times 3>300$, these cannot all be the same foot, so without loss of generality we can assume that there are two right-foot majorities and one left-foot majority. Now the number of right boots in the right-foot majority sizes is at most $300$, so there are at least $100$ left boots in those two sizes, which can all be matched to right boots, giving us $100$ usable pairs as required.