The already correct answers can be summarized in one statement:
"The speed of light is really fast."
But to be more quantitative, we all know Wheeler's famous: "Matter tells spacetime how to curve, and spacetime tells matter how to move."
The first part of that statement is captured in Einstein's field equations (which I am not going to look up, so):
$$ G_{\mu\nu} \propto T_{\mu\nu} $$
The LHS is a curvature tensor, and the RHS is the stress-energy tensor.
Now for a mass in motion, all we have is four momentum:
$$ p_{\mu} = (E/c, \vec p) = \gamma m(c, \vec v) $$
Note that the time component, $p_0$, is huge: $c$, while the space component is an ordinary velocity, perhaps zero.
From that 4-vector, we can construct two symmetric 4-tensors:
$$ m^2\eta_{\mu\nu}, p_{\mu}p_{nu} $$
The stress energy tensor is related to the latter. So at rest, all that is non-zero is the time-time component
$$p_0^2 = (E/c)^2 = (mc^2)^2/c^2 = (mc)^2 $$
Which says: mass causes gravity. Ofc, relativity says pressure, stress, energy flux etc also cause gravity, but mass dominates under normal conditions.
The second 1/2 of Wheeler's statement is captured in the geodesic eq:
$$\frac{d^2 x^{\mu}}{ds^2} + \Gamma^{\mu}_{\alpha\beta}u^{\alpha}u^{\beta}
=0$$
The first term is an acceleration, as in $\vec F = m\vec a$, and the second term I have been a bit lazy and just written a velocity. Roughly:
$$ u_{\mu} = p_{\mu}/c = \gamma(c, \vec v) $$
which is also dominated by the time component, $c$.
At $\vec v = 0 $ in the Schwarzschild metric, this becomes, for $r \gg R_S$:
$$ \ddot r = -\frac{GM}{r^2} $$
which is Newton's law of gravity.
It's a good exercise to work that all out in detail. The result is that all the action is in the time or time-time components of the various 4-vectors and tensors.
Note that the same is true in electromagnetism, where the electric force is much stronger than magnetism. Though it's somewhere unit-system dependent,
the force between two 1C charges at 1 m is:
$$ F = \frac 1 {4\pi\epsilon_0} \frac{(1\,{\rm C})^2}{(1\,{\rm m})^2} \approx 9\,{\rm GN} $$
while the force per meter between two wires carrying 1 amp each separated by a meter is $200\,$ nN.
