A timelike geodesic is the world line for a free-falling object. In the absence of gravity, spacetime is flat (that is Minkowski spacetime, the spacetime equivalent of flat Euclidean space) and the world line of a free-falling object is just a straight line through spacetime, which we would see as the object moving along a straight line in space at a uniform velocity.
But where there is gravity, spacetime is curved, so just as the geodesics on a curved surface are not straight lines, the geodesics through spacetime are no longer straight lines.
For example, the timelike geodesic of the Earth as it orbits the Sun is a kind of elliptical helix through spacetime, as it follows an ellipse in space while its time coordinate increases.
