Basically, you still have only one vote, but you can use that to cast a negative vote against a candidate, rather than a positive one for a candidate.
It's the simplest way to break the monopoly of two parties because a negative vote against a major candidate has the exact same practical effect as a positive vote, but without perpetuating the two-party system. Best of all, those who are afraid of change can vote with a positive vote in exactly the same way they have in the past, with the same effect.
Here's the big problem with that idea: someone still has to win. Therefore you should vote for the candidate who you think deserves to win. If you don't think any of the candidates deserve to win, then that's a problem... but it's not a problem that "negative voting" solves.
IIUC a negative vote would subtract a vote from a candidate instead of adding. In theory, if enough people vote negatively for the two big parties, a third one could arise and perhapas surpass them after gaining enough traction.
Imagine Alice, Bob, and Eve are running for president. My main concern is keeping is keeping Eve out. As for Alice and Bob, they have their pros and cons, but at the end of the day I don't care too much either way and both are much better than Eve. In this scenario a negative vote for Eve would be more in line with my desires than flipping a coin and choosing between voting for Alice or Bob.
Basically, you still have only one vote, but you can use that to cast a negative vote against a candidate, rather than a positive one for a candidate.
It's the simplest way to break the monopoly of two parties because a negative vote against a major candidate has the exact same practical effect as a positive vote, but without perpetuating the two-party system. Best of all, those who are afraid of change can vote with a positive vote in exactly the same way they have in the past, with the same effect.