Newton's first and second laws of motion respectively state momentum is unchanged without a net force and the rate of momentum is equal to net force. Every student naturally wonders what the point is of the first law, as surely it's a special case of the second. But whatever Newton's original motive for such delineation, it's helpful here to understand what geodesics do in general relativity.
Let's first phrase the first law (for a constant mass) another way: without a net force, velocity is fixed, so displacement is a linear function of time. Or, if we construe such a path as through spacetime, we can see its rate of change with respect to time as constant. In other words, it's a straight line through spacetime.
Now, when people say of the GR it implies "gravity isn't a force", what they mean is its equivalent of Newton's first law is a specification of the paths followed in the absence of non-gravitational forces. These paths, which gravity determines, are the geodesics, the equivalent of the aforementioned straight lines. But they're only straight with a suitable parameterization that bears in mind both time and space in the right way, and the metric tensor encodes which way is right.
Explicitly, the first law for a constant mass is upgraded from $\frac{d^2}{dt^2}x^i=0$ to$$\frac{d^2}{d\tau^2}x^\mu+\Gamma^\mu_{\nu\rho}\frac{d}{d\tau}x^\nu\frac{d}{d\tau}x^\rho=0,$$with $\tau$ the proper time. (The metric determines the Christoffel symbols $\Gamma^\mu_{\nu\rho}$.) Either way, the $=0$ assumes no force, and is replaced with $=f^i$ or $=f^\mu$, a force or $4$-force, when we generalize to the second law. But GR's key insight into gravity is that there's another term at work here. Rearranging,$$\ddot{x}^\mu=f^\mu-\Gamma^\mu_{\nu\rho}\dot{x}^\nu\dot{x}^\rho$$(with dots indicating $\tau$-derivatives). The term $-\Gamma^\mu_{\nu\rho}\dot{x}^\nu\dot{x}^\rho$ looks like a $4$-force, which is why objects seem to "fall" and "orbit" and so on. But its origin is fundamentally different.
People always offer a qualitative analogy whereby the spacetime geometry is like bends in a rubber sheet. But I'd like to offer another one. We see from the above that geometry-induced gravity adds something quadratic in $\dot{x}$ to $\ddot{x}$. On at least one model, that's reminiscent of the drag force in a fluid. In this analogy, a spacetime geometry further from "flat" spacetime (i.e. stronger gravity) is like a stronger drag force, while a geodesic is like the path a body with no propulsion (and, ironically, no gravity!) acting on it will follow due to drag. I'm sure this analogy has all sorts of drawbacks on further scrutiny, but it hopefully makes GR's account of gravity seem less bizarre.
