Why is general relativity a geometric theory?

Without getting formal, a geometric theory such as General Relativity (GR) is one where the geometry is a dynamical variable.

In GR the dynamical variables are the entities that define the geometry of the space time. The basic entity is the metric g, a 4x4 tensor (think of it as a 4x4 matrix, except to multiply 2 of them, or one by a 4-vector you have to be careful to have the indices raised or lowered to match the Einstein summation rule. There is a more elegant formulation using differential geometry concepts and notation). The metric is a tensor, but not an invariant, and a different coordinate system will give you a different metric. With the coordinate system, the metric defines distances. There are geometrical entities defined from the metric that determine the invariant spacetime curvature and certain properties. The metric and the geometry is not fixed, it is a variable which depends on the distribution of mass-energy and of course initial or other conditions.

Bottom line, the geometry can change, and it determines how matter and radiation moves in that spacetime. So all the dynamics involve the geometry. Thus a geometrical theory.

Quantum Field Theory (QFT), Special Relativity, and Classical Field Theory are not geometrical theories. The spacetime is defined a priori, and does not change. The spacetime is those cases is simply flat spacetime, also called Lorentz spacetime. Note that QFT is used, with gauge theory, for the electromagnetic, strong and weak forces in physics. There is still no accepted theory that unifies quantum physics with GR, though one can do QFT in a fixed GR spacetime (the spacetime may change with time, but not due to the quantum particles affecting it, so it's ok for small quantum effect in GR, but not for strong gravity and strong interactions).

Classical pre-relativistic physics has only space (with time a separate and independent coordinate which does not mix with spatial coordinates), and they are also non-geometric because neither space not time are dynamical - they represent a fixed background, and do not change. The space is Galilean 3D space.

String Theory, in its various forms is also a geometric theory, with strings and other higher dimensional objects defined in a spacetime which is 10 or 11 dimensions, and with the geometric theory quantized. It's not a proven theory.

There are other geometrical and non geometrical theories. Hopefully this gives you an idea of what is meant.

Your next quest should be to understand why gravity is geometrical and the other forces not, and why it's so hard to combine them.