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
"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.
is the path
- there is no other -
to purify vision.
and that will be Mara's
you put an end
to suffering & stress.
I have taught you this path
- for your knowing -
the extraction of arrows.
It's for you to strive
point out the way.
Those who practice,
absorbed in jhana:
from Mara's bonds
they'll be freed.