Learn German: Ich schreibe die Liste auf, damit du sie hast. - I am writing down the list so that you have it.

The word damit (so that) is a subordinating conjunction. It introduces a dependent clause, which forces the conjugated verb (hast) to move to the very end of the sentence.

Why does it say sie here? Doesn't sie usually mean 'she' or 'they'?

In German, pronouns must match the gender of the noun they replace. Because Liste is a feminine noun (die Liste), we use the feminine pronoun sie to mean 'it' when referring back to the list.

Why is the word auf sitting right before the comma?

The verb in the first part is aufschreiben (to write down), which is a separable verb. In a regular main clause, the prefix (auf) splits off from the main verb (schreibe) and goes to the end of its clause, right before the comma.

Could I use um ... zu instead of damit to say 'so that' here?

Not in this sentence! You can only use um ... zu when the subject is the same in both halves of the sentence. Here, the first subject is ich (I) and the second subject is du (you). When the subjects are different, you must use damit.