It is a teaching tool that shows us how a single consciousness appears to be divided as soon as objects appear within it.

These can be objects such as thoughts and feelings, but also spatial objects. Just as space is seemingly conditioned or limited by multiple pots to form individual pot-spaces (ghaṭākāśa, ghaṭa, pot & ākāśa, space).

Just as space (ākāśa), though seemingly different due to various conditioning adjuncts (upādhis), remains one and omnipresent, all-pervading consciousness (vibhu) appears to be different due to the limiting conditions of the body-mind complex. When these adjuncts are removed, only the one undivided reality remains.

The upādhi teaching fits into the avaccheda model. An apparent factor (upādhi) lends its properties or attributes, for example, colored light, as it were, to the carrier, the upahita, for example, a clear crystal. Similarly, mayā or ignorance seems to lend their attributes to consciousness. In this way, the upahita is seemingly conditioned or limited. Because of this, consciousness appears to be conditioned (cut up, avachinna) by the concepts produced in the mind.

In reality, this is merely upādhi activity (color in clear crystal, the movements of the mind in consciousness), and the experienced consciousness is nothing other than pure consciousness. The following two verses also express this very aptly:

vivekacūḍāmaṇi verse 94 (Ādi Śaṅkara):

yathā ghaṭastho vibhākāśo nīyate ghaṭamārutaiḥ
tathā jīvo vibhā brahmaṇi nīyate karmabhiḥ

"Just as the space inside a pot appears to move when the pot moves, so too does the jīva appear to move due to karma, although he is always one with Brahman."

And dakṣiṇāmūrtistotra verse 4 (Also Ādi Śaṅkara)

bīja syāntari vāṅkuro jagadidaṁ prāṅnirvikalpaṁ punaḥ
māyā kalpita deśa kāla kalanā vaicitrya citrīkṛtam

"Just as a sprouting tree is hidden within a seed,
so this world originally existed in an undifferentiated state.
But due to the illusion of māyā, it is apparently divided into space, time, and diversity."

Māyā appears to divide reality, but after knowledge, it turns out to be undivided.

The avaccheda model keeps jīva (the individual) and īśvara (God, the Lord) separate. Avidyā is the upādhi of the individual, and māyā is the upādhi of īśvara. The limitation of this model lies in the fact that the individual's ignorance is, in reality, a direct consequence of, and therefore part of, māyā. This is explained in the more subtle ābhāsa vāda model.

Avaccheda vāda is often mentioned alongside two other models, highlighted here for their didactic utility:

Avaccheda vāda is used for simple explanation (e.g., "You are Brahman limited by ignorance").
Ābhāsa vāda is used for radical negation (e.g., "The world is mithyā").
Pratibimba vāda is used in devotional contexts; for example, jīva is a reflection of Īśvara.

The Vedānta tradition associated with avaccheda vāda is called bhāmatī prasthāna. See there for more nuances.