I have come across this beautiful illustration of how gravity works both in Newton's framework and Einstein's General Relativity.
In a way, a falling object in General Relativity is obeying Newton's First Law - an object remains at rest or moves in a straight line unless disturbed by a force. A falling object is moving in a straight line, but in curved spacetime. As a result of moving in a straight line it moves towards a source of gravity. Gravity isn't a force - it is a distortion of space and time that changes what it means for an object to travel in a straight line.
You might wonder why an object at rest starts falling in General Relativity if there is no force of gravity. This is because gravity bends not just space, but time as well. The bending mixes up space and time, so that whereas without gravity a stationary object would move only in time and so would stay where it is, with gravity some of the movement in time is switched to a movement in space. So, what can be thought of as the object travelling in a straight line in the time direction (and so stationary) becomes the object travelling in a straight line in a mixture of time and space, and so it falls.