A spinning ball in a fluid generates a force called the Magnus Effect. The direction it spins determines where that lift points. The ball is also being thrown forwards. Just like a satellite orbiting a planet or a rock with a string tied around it, if you apply a force perpendicular to an objects direction of motion, it will begin to move in a circle.

In the very simple case where the curveball's curve axis is perfectly perpendicular to its velocity, it will begin to curve along the arc of a circle. Baseballs are pretty heavy and they're being thrown pretty fast, so the actual break relative to a straight line may only be a few inches but this is the classic curveball description.

Another factor in a late break is slow down due to drag. A baseball can loose around 10% of its forward velocity due to drag forces over the 60ish ft it travels. If the spin of the ball isn't too badly affected, the arc of the ball can dynamically "tighten" the turn near the end of the trajectory simply because the ratio of lift force and forward velocity is changing.

In most cases, the spin axis is not perfectly perpendicular to the velocity. The perpendicular component will still cause it to move like a typical curve, but that other component can be used to have it break laterally or simply hang longer until gravity and/or drag make it break later. If these secondary spin components are small, combined with slowing forward motion, these subtler lift components lift may not become observable until later in the trajectory. In these cases, there is also a degree of batter psychology and surprise that can enhance this effect.