Imagine that you have a perfectly flat, frictionless trampoline. This is a representation of relativistic space-time. If you roll a golf ball across the surface, it will travel in a straight line.
Now imagine that you place a bowling ball in the center of the trampoline. The mass of the bowling ball will create a depression (curvature) in the trampoline. If the golf ball is now rolled across the surface, it will follow the curvature of the surface, and curve around the bowling ball.
A gravitational force between the two balls is not required to change the motion of the golf ball, it simply follows the curvature of the surface.
If the golf ball is rolled too close to the bowling ball, or too slowly past the bowling ball, the curving motion will be greater, possibly even resulting in the golf ball crashing into the bowling ball.
But if the golf ball is rolled farther from the bowling ball, or more quickly past the bowling ball, the curving motion will be less, and the golf ball will continue moving past the bowling ball, although its direction of motion will be altered.