Learn German: Die Enkelkinder, die ich oft besuche, wachsen sehr schnell auf. - The grandchildren that I often visit are growing up very fast.

The second die means "that" or "whom" here. It is a relative pronoun that connects the extra information back to Die Enkelkinder (the grandchildren). Because Enkelkinder is plural, the relative pronoun is die.

Why is the verb besuche at the very end of its section?

The phrase die ich oft besuche is a subordinate clause. Just like the conjunctions seit and seitdem from your current lesson, relative pronouns like die kick the conjugated verb (besuche) to the very end of their clause.

Why doesn't the verb wachsen come immediately after the first comma?

The main sentence is actually Die Enkelkinder wachsen sehr schnell auf. The relative clause (die ich oft besuche) is just a descriptive interruption inserted into the middle of it. So wachsen stays in its normal spot directly after the full subject.

Why is auf sitting all by itself at the very end of the sentence?

It belongs to the separable verb aufwachsen (to grow up). In a regular main clause, the conjugated part of the verb (wachsen) goes in the second position, and its prefix (auf) gets sent to the very end of the sentence.