When something is exact, it's on the dot. When it's 2:00 p.m. on the dot, it is precisely 2:00 p.m. — not a minute earlier or later.
A dot is very small. If you tried to hit a dot with a tiny dart, you'd have to be incredibly accurate. Similarly, anything on the dot is done precisely. The expression on the dot usually refers to an exact time. If a meeting takes an hour on the dot, it didn't go one second over. This term is all about precision.