Don’t think of the metric tensor as just a “multiplication factor”, or as something to raise and lower tensor indices. Think of it as what determines distance in spacetime. It’s the coefficients of the terms in the Pythagorean Theorem!

For example, in flat spacetime (and taking $c=1$) the four-dimensional distance between nearby points is

$$ds^2=dx^2+dy^2+dz^2-dt^2$$

but at a particular point in a curved spacetime it could be something like

$$ds^2=1.12\,dx^2+0.05\,dx dy+0.97\,dy^2+1.27\,dz^2-0.85\,dt^2.$$

In General Relativity, particles free of non-gravitational forces move on geodesics through curved spacetime. Geodesics are paths of minimum/maximum/stationary distance, so knowing the metric tensor means you can find out how things move under gravity.

Spacetime is curved by the density and flow of energy and momentum. The Einstein field equations specify how the energy-momentum tensor determines the metric tensor.

So the “big picture” is: energy and momentum cause distance in spacetime to be more complicated than $ds^2=dx^2+dy^2+dz^2-dt^2$. In other words, they cause a non-Minkowskan metric tensor. This curved geometry then causes the “straightest possible lines” in it to be non-trivial gravitational trajectories (including Earth’s elliptical orbit!).