How does the vice president assist with passing a law?
The vice president typically has very little to do with a law passing.
In formal terms, the vice president can only vote on a law if the Senate vote on that law is tied. In that case, the vice president casts the tie-breaking vote.
Vice presidents can have an informal impact on making laws. Vice presidents may help the president try to lobby members of Congress to vote for a law. This does not always happen and it is not a formal power of the vice president.
In terms of the formal rules of “how a bill becomes a law,” then, the vice president has little to do with a law passing.