People use this expression when they're trying to do something at the same time. First you say this phrase, then you count "One, two, three". When you say the word "three", everyone does the action at the same time. Here are some other situations where you can use "On the count of three":
- When some people are all lifting something heavy together.
- When you and a friend are each showing something embarrassing to each other.
People usually don't use "On the count of three" at the beginning of a race, by the way. To start a race, people say:
On your marks. Get set. Go!
You could use "On the count of five" or some other number, but English speakers mostly use "three".