He is the president of the Senate. There is no conflict of interest as a result. Although the slime-o-crats will howl like hell because of it.
I think he meant that the conflict of interest is that if Trump is convicted and removed, Pence becomes president. Pence stands to gain by voting with the Democrats. (Not saying he would. Besides, if the Democrats don't have the votes to prevent a dismissal without a tie-breaker vote from the VP, they certainly don't have them for a conviction.)
Another way to look at this is, what if they had impeached Pence? Would he still get to be the tie-breaking vote in the Senate dismissal vote? That I do not know.