Learn Hardcore Icelandic: Ég fer nú heim.

Because fara is the infinitive form, meaning to go. After the subject ég (I), Icelandic uses the 1st person singular present tense form:

að fara = to go
ég fer = I go / I am going

So Ég fer nú heim uses the correctly conjugated verb form for I.

What exactly does nú mean here?

Nú usually means now, but in actual speech it can be a little flexible.

In this sentence, it most naturally adds the idea of:

now
at this point
right now

So it can sound like the speaker is deciding or announcing the action at this moment: I’m going home now.

In some contexts, nú can also have a discourse flavor, a bit like well, now or so, now, but here now is the most straightforward understanding.

Why is it heim and not heima?