Discipula aperte dicit se mendacio non credere et testem audire velle.

Because discipula is the subject of the main verb dicit (the student says). In Latin, the subject of a finite verb is normally in the nominative case.

Aperte is an adverb (from the adjective apertus, -a, -um, meaning open/clear). It modifies dicit and tells you how she speaks: she says openly/clearly.

Because this is an indirect statement after dicit, and se is the reflexive pronoun used to refer back to the subject of the reporting verb (discipula).
So se = herself = the student (not some other female person).

It’s an indirect statement (often called accusative + infinitive). After verbs of saying/thinking/knowing, Latin commonly expresses that... with:

• an accusative subject: se
• an infinitive verb: credere

So structurally it’s: she says [that] she does not believe...

Because credo normally takes the dative for the person/thing believed. So:

• mendacio credere = to believe a lie (literally to give belief to a lie)
  That’s why it’s mendacio (dative singular), not mendacium (accusative).

Non negates the infinitive credere: not to believe. In context, it negates the first infinitive clause introduced by se: se mendacio non credere.

Because et coordinates two things the student says about herself: 1) se mendacio non credere
2) (se) testem audire velle

The second one doesn’t repeat se, but it’s understood from the first clause.

Because audio takes a direct object in the accusative: testem audire = to hear the witness.
So testem is the object of audire.

Volo/velle is commonly followed by a complementary infinitive that completes its meaning:

• velle audire / audire velle = to want to hear

Both word orders are possible; Latin can place either infinitive first depending on emphasis and style.