The statement that curved space causes gravity is a simplified way of expressing the relationship between the curvature of spacetime and the effects of gravity. In the theory of general relativity, developed by Albert Einstein, gravity is described as the curvature of spacetime caused by the presence of mass and energy.
According to general relativity, mass and energy not only affect objects by exerting forces on them, but they also curve the fabric of spacetime itself. This curvature is what we perceive as the force of gravity. When a massive object, such as a planet or a star, is present in spacetime, it causes a curvature in its vicinity. The curvature of spacetime influences the motion of other objects, causing them to move along curved paths or orbits around the massive object.
In this sense, the presence of mass and energy creates a gravitational field that determines the paths of objects moving through spacetime. The objects themselves follow these curved paths because they are moving in a region of curved spacetime. This effect is often visualized using the analogy of a mass placed on a rubber sheet, causing the sheet to deform and objects placed on it to move towards the mass.
So, while it is true that mass causes objects to move along curved paths through space, the underlying reason for this motion is the curvature of spacetime caused by the presence of mass and energy. The statement that curved space causes gravity is a shorthand way of expressing this relationship in the context of general relativity.