Here's why:
* Loudly is an adverb that modifies the verb "singing." It describes *how* the person is singing.
* Loud can be used as an adjective or an adverb, but in this case, it would be an adjective describing the *volume* of the singing, which is not the intended meaning.
The sentence "I am not singing that loud" would mean that the person's singing is not *loud* in nature, rather than the volume being low.