The Buddha explained it in terms of a threefold knowledge (tevijja).
When He was sat alone under the Bodhi Tree during the full moon in May (Vesak), he came to know 1) His previous lives 2) the future destinations (after death) of others, and 3) The Four Noble Truths.
This last knowledge, or Bodhi ~Naa.na included the realisation of Nirvaana for Him, the Highest Heaven, total escape from the realm of rebirth, wandering on.
He knew that His cankers had been destroyed, and there was no more for Him to do (for Himself) in this life.
Then began His Ministry, and He was happy to find that His Enlightenment experience was communicable on various levels.
The Highest being the equalling of Himself in another, in Arahantship.
That's all I can think of to say right now; thank you for your question. : )
MA in Buddhist Studies.