Why does the Buddha call himself the Tathāgata?

I cannot seem to find in which Sutta this is said, but... The Blessed One said once: Bhikkhus, the world has been discovered by the Perfect One, he has dissociated from the world. The origin of the world has been discovered by Him. He has abandoned the origin of the world. The cessation of the world has been discovered and He has realised the truth. In the world with its deities and others, whatever can be seen, heard sensed and cognized or reached, sought out and encompassed by the mind, has been discovered by the Perfect One that is why He is called a Perfect One (The Thathagatha). The Most Enlightened.

Unrul3r's answer includes that it might mean going or coming (gata or agata).

Andrei's answer is that it means 'grounded' or 'rooted' (which makes sense to me: because if you go or come, have gone or have come, to somewhere then that is where you are).

Combined (and further to user319's answer) these remind me of a Zen story -- using Google I cannot find a reference to it at the moment, so I paraphrase it:

Someone, who is dying, says, "We come from nowhere, and we go to nowhere."

His companion, to be helpful, replies, "Let me show you the road, where there is no 'coming' and no 'going'."

Tathagata is one who will not be reborn. In the Sutta the word is used only for Sammasambuddha. It cannot be separated as Tatha + Agata; or Tatha + Gata. It is Tathaa + gata.

A Tathagata has stopped 'moving' is Sansara.

My guess for this question is Tathāgata means "arrived at/come to the truth/reality".

The Pali Dictionary say the origin & etymology of Tathāgata is unknown, including to Buddhaghosa:

Derivation uncertain. Buddhaghosa (DN-a.i.59–DN-a.i.67) gives eight explanations showing that there was no fixed tradition on the point, and that he himself was in doubt.

It appears there are two possible derivations:

• tatha + āgata = reality/truth come/arrived
• tathā + gata = thus gone

The seemingly unambiguous contextual non-Buddha use of the term in AN 6.54 seems to explain the term, as follows:

“Sir, I’ve been banished by the local lay followers from all seven monasteries in my native land.”

“Pabbājito ahaṁ, bhante, jātibhūmakehi upāsakehi sabbaso jātibhūmiyaṁ sattahi āvāsehī”ti.

“Enough, Brahmin Dhammika, what’s that to you? Now that you’ve been banished from all of those places, you have come to me.

“Alaṁ, brāhmaṇa dhammika, kiṁ te iminā, yaṁ taṁ tato tato pabbājenti, so tvaṁ tato tato pabbājito mameva santike āgacchasi.

Once upon a time, some sea-merchants set sail for the ocean deeps, taking with them a land-spotting bird.

Bhūtapubbaṁ, brāhmaṇa dhammika, sāmuddikā vāṇijā tīradassiṁ sakuṇaṁ gahetvā nāvāya samuddaṁ ajjhogāhanti.

When their ship was out of sight of land, they released the bird.

Te atīradakkhiṇiyā nāvāya tīradassiṁ sakuṇaṁ muñcanti.

It flew right away to the east, the west, the north, the south, upwards, and in-between.

So gacchateva puratthimaṁ disaṁ, gacchati pacchimaṁ disaṁ, gacchati uttaraṁ disaṁ, gacchati dakkhiṇaṁ disaṁ, gacchati uddhaṁ, gacchati anudisaṁ.

If it saw land on any side, it went [arrived] there and stayed.

Sace [if] so [he; the bird] samantā [everywhere] tīraṁ [shore] passati [sees], tathāgatako [truely arriver/comer] va hoti [is a].

But if it saw no land on any side it returned to the ship.

Sace pana so samantā tīraṁ na passati tameva nāvaṁ paccāgacchati.

In the same way, now that you’ve been banished from all of those places, you have come to me.

Evamevaṁ kho, brāhmaṇa dhammika, yaṁ taṁ tato tato pabbājenti so tvaṁ tato tato pabbājito mameva santike āgacchasi.

āgacchati = comes, comes near; comes back, returns; reaches, attains, arrives

pp āgata

Alternately, the independent Australian translator Sujato on Sutta Central asserted the use of "tathāgata" in AN 6.54 is unrelated to the use of tathāgata as the Buddha's name. The impression is, for Sujato, tathāgata as the Buddha's name means "arrived/come to the truth" but in AN 6.54 tathāgata means this bird has "thus gone/flown". Sujato said:

Oh, and to forestall the obvious, no this doesn’t have anything to do with tathāgata as an epithet of the Buddha. The terms in this compound are extremely common and occur in a wide variety of senses. As an epithet of the Buddha, the primary sense is “one who has realized the truth”. How to translate ‘tathāgatakova’?

To me, it seems plainly obvious the analogy in AN 6.54 logically is about the bird & thus the ship arriving/coming to land, just as the ex-Brahmin had come to the Buddha. There is the word 'āgacchati' about the ex-Brahmin coming to the Buddha and the past participle of āgacchati, namely, āgata, in the term 'tathāgatakova' about the bird/ship coming/arriving to land.

To conclude, for me, Tathagata means 'come to the truth'; derived from tatha + āgata.

Thus Iti 112 says, per Sujato translation:

In this world—with its gods, Māras, and Brahmās, this population with its ascetics and brahmins, its gods and humans—whatever is seen, heard, thought, known, attained, sought, and explored by the mind, all that has been understood by the Realized One.

Yaṁ, bhikkhave, sadevakassa lokassa samārakassa sabrahmakassa sassamaṇabrāhmaṇiyā pajāya sadevamanussāya diṭṭhaṁ sutaṁ mutaṁ viññātaṁ pattaṁ pariyesitaṁ anuvicaritaṁ manasā yasmā taṁ tathāgatena abhisambuddhaṁ,

That’s why he’s called the ‘Realized One’.

tasmā ‘tathāgato’ti vuccati.

From the night when the Realized One understands the supreme perfect awakening until the night he becomes fully extinguished—in the element of extinguishment with nothing left over—everything he speaks, says, and expresses is real, not otherwise.

Yañca, bhikkhave, rattiṁ tathāgato anuttaraṁ sammāsambodhiṁ abhisambujjhati, yañca rattiṁ anupādisesāya nibbānadhātuyā parinibbāyati, yaṁ etasmiṁ antare bhāsati lapati niddisati, sabbaṁ taṁ tatheva hoti no aññathā,

That’s why he’s called the ‘Realized One’.

tasmā ‘tathāgato’ti vuccati.

The Realized One does as he says, and says as he does.

Yathāvādī, bhikkhave, tathāgato tathākārī, yathākārī tathāvādī,

Since this is so,

iti yathāvādī tathākārī yathākārī tathāvādī,

that’s why he’s called the ‘Realized One’.

tasmā ‘tathāgato’ti vuccati.

tathāgata

tatha: (being) in truth，truthful； true，real

ā: to，towards

gata: directed to

*canonics

A Tathagata is one who is neither coming or going. That is neither going to the other side or coming back. Because there are no longer sides. Just what is.

1. Why does the Buddha call him self Tathāgata?

What would be better than showing the man's own words?

"Whatever in this world — with its devas, Maras, & Brahmas, its generations complete with contemplatives & brahmans, princes & men — is seen, heard, sensed, cognized, attained, sought after, pondered by the intellect, that has been fully awakened to by the Tathāgata. Thus he is called the Tathāgata.

"From the night the Tathāgata fully awakens to the unsurpassed Right Self-awakening to the night he is totally unbound in the Unbinding property with no fuel remaining, whatever the Tathāgata has said, spoken, explained is just so (tathā) and not otherwise. Thus he is called the Tathāgata.

"The Tathāgata is one who does in line with (tathā) what he teaches, one who teaches in line with (tathā) what he does. Thus he is called the Tathāgata.

"In this world with its devas, Maras, & Brahmas, its generations complete with contemplatives & brahmans, princes & men, the Tathāgata is the unconquered conqueror, all-seeing, the wielder of power. Thus he is called the Tathāgata.
-Iti 4.13, Discourse on the World (Lokasuttaṃ)

In addition to the above, DN 29 provides one more definition:

'If "the past" refers to what is not factual, not true (atathā), to what is not of advantage, the Tathāgata makes no reply. If it refers to what is factual, true (tathā), but which is not of advantage, the Tathāgata makes no reply. But if "the past" refers to what is factual, true (tathā), and which is of advantage, then the Tathāgata knows the right time to reply. The same applies to the future and the present. Therefore, Cunda, the Tathāgata is called the one who declares the time, the fact, the advantage, the Dhamma and the discipline. That is why he is called Tathāgata.'
-DN 29, Delightful Discourse (Pāsādika-suttaṃ)

2. What is the significance of the term?

As the Wikipedia states:

Modern scholarly opinion generally opines that Sanskrit grammar offers at least two possibilities for breaking up the compound word: either tathā and āgata or tathā and gata. Tathā means thus in Sanskrit and Pali, and Buddhist thought takes this to refer to what is called reality as-it-is (yathā-bhūta). This reality is also referred to as thusness or suchness (tathatā) indicating simply that it (reality) is what it is.

A Buddha or an arahant is defined as someone who "knows and sees reality as-it-is" (yathā bhūta ñāna dassana). Gata "gone" is the past passive participle of the verbal root gam "go, travel". Āgata "come" is the past passive participle of the verb meaning "come, arrive".

Thus in this interpretation Tathāgata means literally either “the one who has gone to suchness” or "the one who has arrived at suchness".
-Tathāgata Wikipedia Page

So it appears these two meanings can be derived from the word. I think some light can be shed by seeing the word used in another context in which there can only be one meaning due to the nature of it's context (simile):

In ancient times when seafaring merchants put to sea in ships, they took with them a bird to sight land. When the ship was out of sight of land, they released the bird; and it flew eastward and westward, northward and southward, upward and all around. And if the bird saw no land, it returned to the ship; but if the bird sighted land nearby, it was truly gone (tathāgatakova).
-AN 6.54, Discourse about Dhammika (Dhammika-suttaṃ)

Note that footnote nº2 of the above discourse is also worth reading to get a better grip on it's meaning.

3. How is the term used by the Buddha in the suttas?

As user70 pointed out, the Buddha usually uses the term to refer to himself. Above, I quoted a number of prosaic contexts that reveal how he uses the term, so I'll leave a poetic reference for this answer.

    Just this
    is the path
 - there is no other -
to purify vision.
    Follow it,
and that will be Mara's
    bewilderment.

Following it,
you put an end
to suffering & stress.
I have taught you this path
having known
 - for your knowing -
the extraction of arrows.

It's for you to strive
    ardently.
Tathāgatas simply
point out the way.
Those who practice,
absorbed in jhana:
    from Mara's bonds
    they'll be freed.

-Dhp v275-276

From an academic perspective the wikipedia article "Tathāgata" has good information. According to wikipedia:

The word's original significance is not known and there has been speculation about it since at least the time of Buddhaghosa, who gives eight interpretations of the word, each with different etymological support, in his commentary on the Digha Nikaya, the Sumangalailasini:

1. He who has arrived in such fashion, i.e. who has worked his way upwards to perfection for the world's good in the same fashion as all previous Buddhas.

2. He who walked in such fashion, i.e. (a) he who at birth took the seven equal steps in the same fashion as all previous Buddhas or (b) he who in the same way as all previous Buddhas went his way to Buddhahood through the four Jhanas and the Paths.

3. He who by the path of knowledge has come at the real essentials of things.

4. He who has won Truth.

5. He who has discerned Truth.

6. He who declares Truth.

7. He whose words and deeds accord.

8. The great physician whose medicine is all-potent.

Modern scholarly opinion generally opines that Sanskrit grammar offers at least two possibilities for breaking up the compound word: either tathā and āgata or tathā and gata. Tathā means thus in Sanskrit and Pali, and Buddhist thought takes this to refer to what is called reality as-it-is (yathā-bhūta). This reality is also referred to as thusness or suchness (tathatā) indicating simply that it (reality) is what it is.

A Buddha or an arahant is defined as someone who "knows and sees reality as-it-is" (yathā bhūta ñāna dassana). Gata "gone" is the past passive participle of the verbal root gam "go, travel". Āgata "come" is the past passive participle of the verb meaning "come, arrive".

Thus in this interpretation Tathāgata means literally either “the one who has gone to suchness” or "the one who has arrived at suchness".

Another interpretation, proposed by the scholar Richard Gombrich, is based on the fact that that, when used as a suffix in compounds, -gata often loses its literal meaning and signifies instead "being". Tathāgata would thus mean "one like that", with no motion in either direction.

According to Theodore Stcherbatsky, the term Tathagata has a non-Buddhist origin, and is best understood when compared to its usage in non-Buddhist works, such as the Mahabharata. Stcherbatsky gives the following example from the Mahabharata:

Just as the footprints of birds (flying) in the sky and fish (swimming) in water cannot be seen, Thus (tātha) is going (gati) of those who have realized the Truth.

Buddha uses the term to refer to himself rather than saying, I, me or myself.

Tatha means "truth", "reality" or literally "so", "such".

gata is often translated as gone, however from my research it looks like it is a suffix that means "firmly grounded in" or "rooted in" -- as in kayagata smrti "mindfulness rooted in the body".

Together this would make Tathagata mean "Firmly established in reality".

I personally associate this word with two other terms: sugata and duggata.

Sugata is a person who fared well, a lucky one, and therefore an optimist.
Duggata is an unfortunate person, a loser, and therefore a pessimist.
Tathagata would then mean neither a pessimist nor an optimist, but realist.