ausziehen
to move out
To permanently leave an apartment, house, or office in order to live or work somewhere else.
Meine Nachbarn ziehen nächste Woche aus.
My neighbors are moving out next week.

to move out / take off
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to move out
To permanently leave an apartment, house, or office in order to live or work somewhere else.
Meine Nachbarn ziehen nächste Woche aus.
My neighbors are moving out next week.

'Ausziehen' is a separable verb. The prefix 'aus-' comes at the end of the sentence in a main clause. In subordinate clauses or in the infinitive with 'zu', the verb stays together.
Er zieht seine Jacke aus.
He takes off his jacket.
The basic meaning is 'etwas herausziehen' (to pull something out). This logic connects all the meanings: you 'pull yourself' out of an apartment (move out), you pull clothing off the body (take off clothes), or you pull out part of a piece of furniture (make it larger).
Er zog den Schlüssel aus dem Schloss.
He pulled the key out of the lock.
The exact meaning becomes clear from the object or the grammar. An item of clothing as the object means 'to take off'. A piece of furniture as the object means 'to extend'. The reflexive pronoun 'sich' means 'to undress'. Without an object in the context of an apartment, it means 'to move out'.
Zieh bitte deine Schuhe aus!
Please take off your shoes!
When saying where someone is moving out from, you use the preposition 'aus' with the dative. This structure is very common.
Sie ist letzten Monat aus ihrer alten Wohnung ausgezogen.
She moved out of her old apartment last month.
There is a difference: 'sich ausziehen' usually means to take off all one's clothes. 'Etwas ausziehen' (with an accusative object) means to take off only a specific item of clothing.
Er zog sich komplett aus, um zu duschen.
He undressed completely in order to shower.
to leave / rely
'verlassen' and 'ausziehen' can both have to do with leaving an apartment, a house, or an office. 'verlassen' only says that you leave; this can be for a short time or forever. 'ausziehen' is more specific: you give up the apartment or house and move to another place. If someone only leaves the house briefly, you do not say 'ausziehen'. When it is about moving, 'ausziehen' is more natural.

Meine Nachbarin zieht Ende des Monats aus ihrer alten Wohnung aus.
My neighbor is moving out of her old apartment at the end of the month.
Bitte verlasse die Wohnung leise, weil das Baby schläft.
Please leave the apartment quietly because the baby is sleeping.