Third parties don't have to win the presidency. They don't have to win a majority in congress.
They just have to win enough senators and representatives so that neither of the two other parties (Democratic and Republican) have a clear majority.
Assuming the Dem's and Repub's are diametrically opposed and in lock step along party lines, then without a majority, neither could effectively govern: assume committee chairs, pass legislation, pass budgets, approve nominations.
The third party (or parties) in sufficient numbers to diminish the majority numbers of the other two could then through their weight around obtaining concessions that are more in tune with the views and values of the American people (that all represent). "We'll give you enough votes to get this legislation passed, providing you give us this committee chair (or whatever)". The third party could regularly swing between siding with the two other parties depending on what was going on, really throwing a wrench into assumed order of things, who chairs what, what legislation makes it out of committee, etc.