Transform or Transforms?

The sentence as written is not unconditionally ungrammatical, but it is only grammatical when it is understood as expressing an unusual idea that you are unlikely to want to express.

"It can melt away anger and transforms your bad mood into a good mood" uses the conjunction and to coordinate the two finite verb phrases "can melt away anger" and "transforms your bad mood into a good mood". This means that in this sentence, the auxiliary verb "can" (which expresses a sense of possibility or uncertainty) goes with only the verb "melt".

But it doesn't make much sense to only have uncertainty about the first part of the sentence. As written, the sentence seems to mean "It's possible that it will melt away anger. Also, it will transform your bad mood into a good mood (no matter what)." That is, it seems to imply that maybe, your bad mood will transform into a good mood but your anger will not melt away. But it seems more likely that what you really want to say is that "It's possible that it will melt away anger. Also, it's possible that it will transform your bad mood into a good mood."

In order to ensure that the scope of the auxiliary can covers both parts of the sentence, you must put the verb tranform into the infinitive form. The resulting sentence ("It can melt away anger and transform your bad mood into a good mood") uses the conjunction and to coordinate the two non-finite verb phrases "melt away anger" and "transform your good mood into a bad mood". According to comments by BillJ, in this construction the auxiliary verb can governs both verb phrases, so both melt and tranform must take the infinitive form.

The sentence as written isn't correct.

Depending on what is being communicated, there are two possible ways of rephrasing it:

It can melt away anger and ^{it can}transform your bad mood into a good mood.
It can melt away anger, and it transforms your bad mood into a good mood.

While either form is grammatical, I would say the first—with the singular transform—is the more likely and natural of the two. Although it can can be added, its elision is fine.