Learn Hardcore Latin: Servus dicit se opus novum hodie incipere. - The servant says that he is beginning a new task today.

Because servus is the subject of the main verb dicit. In Latin, the subject of a finite verb is typically in the nominative:
Servus dicit = The slave/servant says.

What form is dicit, and what does it tell me?

dicit is 3rd person singular, present indicative active of dīcō, dīcere. It tells you:

the subject is he/she/it (here, servus)
the action is presented as present (he says / is saying)

Why does Latin use se ... incipere instead of a word like quod or a clause with that?

Latin commonly reports speech/thought using an indirect statement construction: accusative + infinitive (often abbreviated AcI).
So instead of he says that he begins..., Latin does:

dicit (he says)
se ... incipere (himself ... to begin)

Why is se used instead of eum or is?

se is the reflexive pronoun used when the person in the indirect statement is the of the main verb.So means the servant says .