How is mass curving spacetime? In the GR Einstein's interpretation was in terms of a deformation of the geometry of the space ending in a curvature of spacetime. 

I've thought that mass could do this by interacting with other mass and causes gravitational waves which makes the space curved. But a prof told me that this isn't the case. For example the gravitational waves caused between the earth and the sun is just a very tiny effect which causes spiraling earth to the sun which is of little effect. 

So mass is establishing the geometry of the space and the time. But the question remains how is mass doing that (curving spacetime)?
In general relativity, the curvature of spacetime is an interpretation of  the model for gravity: Starting with the assumptions of gravity being equivalent to acceleration, and the speed of light being constant, a mathematical framework that describes gravity is developed. The mathematical conclusion of this model is that the distances between points is dependent on the distribution of mass and energy. A simple way of describing this is to say that spacetime is curved.

In GR, the apparent force of gravity is explained by the curvature of spacetime. It is reasonable to ask if there is a more fundamental theory that can explain the curvature. String theory provides one possibility. In string theory it is possible to deduce the existence of a particle that would be a graviton, and the from that the field equations that describe gravity. In other words, string theory can predict exactly the same mathematical formulae that arise from curved spacetime in GR. However, on very small scales, string theory makes a series of corrections to the field theory, suggesting that the curvature of spacetime is actually an approximation, which is valid on the large scale, to a string theory of gravity.