A rotating rigid body has kinetic energy because all of its mass elements are moving. For rotation about a fixed axis, their individual translational kinetic energies combine into

$$K_{\mathrm{rot}}=\frac12I\omega^2.$$

Why moment of inertia appears

A point mass at distance $r$ from the axis moves with speed

$$v=r|\omega|.$$

Its kinetic energy is therefore

$$\frac12m(r\omega)^2 =\frac12(mr^2)\omega^2.$$

Summing this expression over all the masses produces the moment of inertia $I$.

Dependence on angular speed

Rotational kinetic energy depends on the square of angular speed. Doubling $|\omega|$ multiplies the rotational kinetic energy by four.

The sign of $\omega$ does not matter because kinetic energy is scalar.

Dependence on mass distribution

For the same angular speed, a body with more mass located far from the axis has a larger moment of inertia and therefore more rotational kinetic energy.

The expression

$$K_{\mathrm{rot}}=\frac12I\omega^2$$

is the rotational counterpart of

$$K=\frac12mv^2,$$

with moment of inertia replacing mass and angular speed replacing linear speed.