In literal sense:
"Om" is the Sound symbol of God. If you want to express God in the form of sound, you should utter this mystical sound "Om". In Bhagwat Gita also God says that:
"maharsinam bhrgur aham
giram asmy ekam aksharam
yajnanam japa-yajno ’smi
sthavaranam himalayah" (Bhagwat Gita: Chapter Ten verse 25)
"Sri Krishna said: Of the great sages I am Bhrgu; Of vibrations / sounds I am the transcendental Om. Of sacrifices I am the chanting of the holy names [japa], and of immovable things I am the Himalayas."
"Tat" means ‘that’; it also refers to God only.
"Sat" means ‘truth’ & God is truth which is everlasting and unchanging.
So "Om Tat Sat" means; what is everlasting and unchanging is God, whose infinite Existence - Consciousness Bliss is denoted by Om. Within Om is everything. Om contains all. Om is all. Om is present everywhere, like light.
As per the Bible which refers to "Om" as "Word":
"In the beginning was the Word, and the Word was with God, and the Word was God." which is simply that "In the beginning was the Om, and the Om was with God, and the Om was God."
So If anyone knows the meaning of the above saying, they need not go anywhere. They know everything.
In Christianity also three words are referred & those three words are "Father, Son, Holy Ghost". These three words of Christianity corresponds to the above said three words only "Sat, Tat, Om". "Father" corresponds to "Sat". "Son", corresponds to "Tat" & "Holy Ghost" corresponds to "Om". These three fold nature of God in Hinduism & Christianity is beautifully related in his book "Autobiography of a Yogi" by great saint Paramhansa Yogananda. As per him:
"Sat" or "Father" aspect of God is one who lives beyond this vibratory creation of ours.
"Tat" or "Son or Christ Consciousness" aspect of God is one who lives within this vibratory creation of ours & is present everywhere within this space or vibratory creation & is called omnipresent God.
"Om" or "Holy Ghost" aspect of God is one who is witness of this "Tat" or "Son or Christ Consciousness" aspect of God.
As per "Paramhansa Yogananda":
The outward manifestation of the omnipresent Christ Consciousness (Tat aspect of God), "It's Witness", is Om, the Word or Holy Ghost. This Om is the invisible divine power, the only doer, the sole causative & activating force that upholds all creation through vibration. Om, the Word is the creative vibration that externalizes all creation.
Brahma - Vishnu - Shiva represents the triune expression of God in the aspect of "Tat or Son", the Christ Consciousness immanent in vibratory creation. The Shaktis, Energies or the "Consorts" of the "Trinity" are symbols of "Om" or the "Holy Ghost" the sole causative force that upholds the Cosmos through vibration.
Below are some verses from Bhagwat Gita also which illustrates the "Om Tat Sat":
"om tat sat iti nirdeso
brahmanas tri-vidhah smrtah
brahmanas tena vedas cha
yajnas cha vihitah pura" (Bhagwat Gita: Chapter Seventeen verse 23)
"Sri Krishna said: O Arjuna, From the beginning of creation, the three words "Om Tat Sat" were used to indicate the Absolute Truth. These three symbolic representations were used by Brahmins while chanting the hymns of the Vedas and during sacrifices for the satisfaction of the God."
"tasmad om ity udahrtya
satatam brahma-vadinam" (Bhagwat Gita: Chapter Seventeen verse 24)
"Sri Krishna said: O Arjuna, Therefore, transcendentalists undertaking performances of sacrifice, charity and penance in accordance with scriptural regulations begin always with Om, to attain the God."
"tad ity anabhisandhaya
dana-kriyas cha vividhah
(Bhagwat Gita: Chapter Seventeen verse 25)
"Sri Krishna said: O Arjuna, Without desiring fruit producing results, one should perform various kinds of sacrifice, penance and charity with the word Tat. The purpose of such transcendental activities is to get free from material entanglement."
"sad-bhave sadhu-bhave cha
sad ity etat prayujyate
prasaste karmani tatha
sac-chabdah partha yujyate"
"yajne tapasi dane cha
sthitih sad iti cocyate
karma chaiva tad-arthiyam
sad ity evabhidhiyate"
(Bhagwat Gita: Chapter Seventeen verse 26-27)
"Sri Krishna said: O Arjuna, The Absolute Truth is the objective of devotional sacrifice, and it is indicated by the word Sat. The performer of such sacrifice is also called Sat, as are all works of sacrifice, penance and charity which, true to the absolute nature, are performed to please the God."