What muscles does the Back Extension work?

Mainly the lower back. It also works the glutes and hamstrings.

What equipment do you need for the Back Extension?

None. It is a bodyweight exercise.

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A supported hip hinge on a 45-degree bench that trains the spinal erectors, glutes and hamstrings with no added load.

Set the hip pad of a 45-degree back extension bench just below your pelvis.
Step onto the platform and lock both ankles under the roller pad with your legs straight.
Cross your arms over your chest, brace your abdominals, and align your head, spine and hips.
Hinge at the hips and lower your torso until you feel the hamstrings lengthen.
Squeeze the glutes and extend the hips until your body forms a straight line.
Hold the top position briefly, lower under control, and repeat for the prescribed reps.

Arching well past a straight line at the top, which loads the lumbar spine instead of the hips.
Setting the hip pad too high, which blocks hip flexion and pushes the movement into the lower back.
Swinging through reps with momentum, which removes tension from the glutes and hamstrings.
Throwing the head back or tucking the chin hard, which breaks the neutral line the exercise trains.

What muscles does the Back Extension work?: Mainly the lower back. It also works the glutes and hamstrings.
What equipment do you need for the Back Extension?: None. It is a bodyweight exercise.
Is the Back Extension good for beginners?: GymVim rates it beginner.