Learn Hardcore Latin: Rubor in cute interdum manet, etiam cum dolor iam minor est. - Redness sometimes remains on the skin, even when the pain is already less.

Because rubor is the subject of manet. The sentence is saying that the redness remains, so rubor is the thing doing the action of the verb.

rubor = redness
manet = remains

So rubor must be nominative singular.

Why is it in cute and not in cutem?

Because in can take either the ablative or the accusative, depending on meaning.

in + ablative = in / on a place, with the idea of location
in + accusative = into / onto a place, with the idea of motion toward

Here the meaning is location: the redness remains in the skin. So Latin uses the ablative:

cute = ablative singular of cutis

If the sentence were about movement into the skin, then in cutem would make sense instead.

What form is cute, and what noun does it come from?

Cute is the ablative singular of cutis, cutis, a feminine third-declension noun meaning skin.

Its basic forms are:

nominative singular: cutis