Who enters Nibbāna, Happy, Free
&
in Peace?
Wrong question!
Why? There was no-one really there in the first place...!
Nibbāna
is there. Beings attain it, but no-one is ever seen entering it...
There was no 'entity' or 'person' there, but a
process that
simply stops!

Right question: Not Who, but What is then
absolutely free?
Right answer: Form is then free, Feeling is
then free, Perception is then free,
Mental Construction is then free, and Consciousness itself is then free!
Cut short: The
5 clusters of clinging are instantly freed!
When all craving and all clinging is completely relinquished and eliminated,
then they are all completely irreversibly freed and they by that evaporate,
disintegrate, fade away and vanish... This is the signless
Nibbāna...
The Blessed Buddha once explained:
When past conditioning is exhausted, and no fresh causality is produced,
then mind is no longer lead to future rebirth. When cravings no more arise,
the seed is consumed! Still-minded ones cease like the flame of this lamp...
Sutta
Nipāta 235
Whenever and wherever one encounters the arising
and passing away of the
mental-physical structure, one enjoys bliss and delight, which lead on to
the deathless stage experienced by the wise.
Dhammapada 374


What actually stops
burning right there?

The Fire
of Greed, Hate and Ignorance, is finally extinguished in
Nibbāna...
On this veritable "Fire
of Painful Existence"
see also:
https://What-Buddha-Said.net/drops/II/Burning.htm
https://What-Buddha-Said.net/drops/II/The_Burning_Five.htm