BAD is an adjective that means “the opposite of good,” “poor quality” or “not well.”
BADLY is the adverb form of “bad.”

Action verbs, which describe an activity or movement, need adverbs to modify how the action is being done.

Linking verbs, which “link” the subject to a predicate, require adjectives to modify them. A linking verb is used to re-identify or describe its subject without expressing an action.

To complicate things, some words can be linking verbs OR action verbs, depending on the sentence. Most of these have to do with the five senses. You can actively “feel” sandpaper or you can “feel” happy. You can “look” closely at something or “look” pretty in a dress.

In the sentence “I feel bad(ly) about that,” the verb FEEL here acts as a linking verb and links the subject to the predicate adjective that describes it. That’s why you need the adjective BAD in that sentence. Thus, you say, “I feel bad about that.”

Some people insist on incorrectly saying: “I feel ‘badly’ about that. However, “badly” is an adverb, not an adjective. If the person really does feel badly (the adverb), it means he has deficient tactile abilities or perhaps a calloused soul incapable of doing a very good job of feeling.

The linking verb “feel” must link noun to adjective, not adverb. The person feels sorry or regretful.

Thus, the correct sentence is, “I feel BAD about that.”