Puer dicit se sitim sentire et lac bibere velle.

Because dicit commonly introduces an indirect statement in Latin. Indirect statements are usually built with:

• an accusative subject (here se), and
• one or more infinitives (here sentire and bibere with velle).

So instead of “He says that …”, Latin often uses “He says himself (acc.) to …”.

Se is the reflexive accusative pronoun, used when the subject of the reporting verb (puer dicit) and the subject of the indirect statement are the same person.

• puer dicit se … = “The boy says that he (himself) …” If it were a different male person, you’d use eum:
• puer dicit eum … = “The boy says that he (someone else) …”

Se is the subject of the infinitives inside the indirect statement. In Latin indirect statement, the subject is put in the accusative:

• se sentire = “that he feels …”
• se … bibere velle = “that he wants to drink …”

Sitis is the dictionary (nominative) form meaning “thirst,” but here it is the direct object of sentire, so it must be accusative:

• sitis (nom.) → sitim (acc.) Latin often says literally “to feel thirst” (sitim sentire).

It’s a common idiom, yes. Another frequent option is the verb sitire (“to be thirsty”), e.g.:

• puer dicit se sitire But your sentence uses the noun + verb pattern:
• se sitim sentire = “to feel thirst / to be thirsty”

Lac is a neuter noun, and neuter nouns have the same form in the nominative and accusative singular. Here it functions as the direct object of bibere, so it is accusative, but it looks identical:

• nom. lac = “milk”
• acc. lac = “milk” (as an object)

Velle (“to want”) is followed by an infinitive expressing what is wanted. So:

• bibere = “to drink”
• velle bibere (or bibere velle) = “to want to drink”

In your sentence, se is the subject of velle, and lac is the object of bibere.

In indirect statement, the infinitive tense is relative to the main verb (dicit):

• present infinitive = action happening at the same time as dicit So the boy is saying (now) that he is thirsty and wants to drink (now).

(If Latin wanted to show earlier or later time, it would typically use perfect or future infinitive constructions.)

Et connects two things inside the same indirect statement—two reported ideas about the same subject (se): 1) se sitim sentire 2) (se) lac bibere velle

Latin often omits repeating se the second time because it’s understood.

Latin word order is flexible, but it often places key verbs—especially the “main” verb of a phrase—toward the end. Here velle is the controlling verb for the second idea (“want”), so it naturally comes last:

• lac bibere velle = “to want to drink milk”

You could also see se velle lac bibere; the meaning is essentially the same, with different emphasis or rhythm.