1. Prajnanam Brahma or Brahman is pure consciousness. (Aitareya Upanishad 3.3)
The Upanishad says Brahman is Full. When fullness is taken out of Fullness, Fullness remains. Fullness refers to Brahman and fullness refers to the jagat or the manifested universe. What is consciousness? The word consciousness is one of the most misunderstood. Consciousness is the subtlest of the substance one can think of. It is subtler than the mind. It is subtler than the thoughts. It is subtler than the space. It is subtler than time. In fact it is beyond space and time. Consciousness it its purest form (non-transferred form) is unchangeable. Something that has no change essentially is beyond space-time modifications. It has permeated the entire unmanifested and manifested worlds. It is that substance that forms the substratum of the entire creation. It is that substance on which the Primordial Nature or Adi Shakti projects its sportive lila. It is the screen on which the Primordial Nature projects its movies on. It is pure existence. It is the absolute Truth or without modification or Sat. Its state is Blissful or Anand or beyond duality. Upanishads declare that Bramhan is Sat-Chit-Ananda or Satchidananda. From Consciousness has evolved everything and every being.
2. Ayam Atma Brahma or This Atman is Brahman. (Mandukya Upanishad 1.2)
This aphorism declares that the Atman or Self or the individual consciousness is same as the universal consciousness. The singular atman is representative of the totality, meaning, the realization with the singular atman is the same realization as that with the Brahman. The nature of both are same. Metaphorically speaking, the electric energy in essence is the same within the fan and the light and displays all its fundamental properties. Knowing the Self is same as knowing the Brahman or the Super Self. There is no further knowledge about the Brahman is obtained beyond what is obtained about the Self. That is why the yogis strive to know the Self.
3. Tat Tvam Asi or Thou art That (Chandogya Upanishad 6.8.7)
This is where the seer is declaring to the disciple, "You are That which is Sat-Chit-Anand, That which is One without a second, One beyond the mind, intellect, and ego, you are pure existence, pure consciousness, and pure bliss". This is later declared by Jesus Christ as "You are made in the image of God". As discussed under Maha Vakya 1, everything is consciousness but the one which is in the innermost recesses of you is in the purest form of consciousness and that is the image of God that Lord Jesus refers to.
4. Aham Brahmasmi or I am Brahman (Brhadaranyaka Upanishad 1.4.10)
This is essentially same as the Mahavakya 3. Here the "I" refers to the Self or the higher ego. This declares that which is the witness within me of all the thoughts and sensory activities is essentially Brahman or pure Consciousness. The Consciousness is there everywhere and everything is made of Consciousness under the influence of the Primordial Nature of the Brahman or Adi Prakriti. An analogy would be to think of iron getting transformed into a hammer, a nail, a knife, etc under the influence of external energy. So Consciousness is transformed into the multiplicity under the influence of Adi Prakriti or Maha Maya. In this transformation process, some (consciousness is immeasurable but the word some is used in attempt to explain) of the consciousness stays in its purest form even from just like in the material world, primordial hydrogen is still present although some hydrogen was transformed into other matter. This is the knowledge (realization) that yogis are after, knowing which there is nothing further remains to be known.
5. Sarvam Khalvidam Brahma or All this is Brahman (Chandogya Upanishad 3.14.1)
This is not treated as a Mahavakya since the earlier four Mahavakyas cover the Truth in its entirety. Basically this aphorism declares that the entire manifested and unmanifested substance is Brahman including thoughts, speech, matter, energy, intellect, emotions, etc. etc. No further explanation is necessary.
6. Neha Nanasti Kinchana or There is nothing else anywhere. (Chandogya Upanishad)
This is not one of the Maha Vakyas but is relevant to include here. It indirectly declares that Brahman is a continuum of Consciousness, without any void. Brahman is the superset whose complementary is null. There is nothing but Brahman everywhere. The multiplicity including matter, energy, and thoughts are mere transformation or projection of Chit or Consciousness. Rajjwa bhujangama eva pratibhasitam vai. The world is merely a notion projected on the Brahman similar to the projection of an idea of a snake on a rope. Brahmarpanam Brahma havir Brahmagnau Brhmanahutam. Brahmaiva tena gantavyam Brahma karma samadhina (Bhagavad Gita). This shloka in Gita clearly indicates that matter, energy, and thoughts are all Brahman.