[Inziladun]

What Gandalf and Tom discussed is to me bound with the question of what Tom was. Bombadil's apparent uniqueness in the legendarium makes speculation especially difficult.
At the moment (because my thoughts on Tom's nature have varied), I'm of the mind that Tom could have been one of the Ainur who entered the world at a different time than the Valar (like Ungoliant?), and had his own 'assignment' from Eru that kept him out of most of the histories of the time. That, or like Radagast, he had his own agenda.
If that scenario is correct, I would say perhaps Gandalf needed someone to open his mind to, discussing all his hopes and fears, his failings and accomplishments; something of a 'confession' which would cleanse whatever weariness or negative feelings he had built up while being confined in flesh for so long. He would then be ready for his last journey homeward.
And what better 'Confessor' than one of the divine beings who had heard the music, but had sat outside as a neutral observer?
That's my theory, anyway.