I find the literal meaning does not fit here.
That is often the case with song lyrics. Figure of speech is heavily used, as is the case here.
OED link. Definitions of the verb:
Mark with a branding iron.
1.1 Mark indelibly.
1.2 Mark out as having a particular bad or shameful quality.
Assign a brand name to.
2.1 The promotion of a particular product or company by means of advertising and distinctive design.
Definition 1 (mark with a branding iron) is correct here, but it is used figuratively. Figuring out the meaning of the lyrics is subjective (and somewhat off topic), but I see two distinct possibilties here:
- He's trying to say that his feet have burn marks/scars on them. Likely because he has been using his feet intensively, and they now hurt.
- He's trying to say that his feet bear the scars of a past event. The rest of the lyrics seem to refer to something that has now disappeared. His branded feet may be a remnant of that which is now lost, in the sense that "being branded" means "having a mark left on you". Somewhat of an analogy (and maybe even exactly what the singer means): There was a party. The party is now over. There is no proof left that there was ever a party, except for the sore feet that the singer has from dancing at the party. His feet bear the scars of having danced at the party.
There is a lot of overlap between these two options. However, option 1 focuses on the existence of the wounds, and option 2 focuses on the fact that the wounds are proof of something in the past.
Because of the figurative usage, it's hard to figure out what exactly is meant with the sentence. But "brand" in and of itself seems clear: visibly marked or scarred.