I've always attributed that to dementia; remember that Bilbo was very old for a hobbit (about 128 years); only the Old Took had been older.
Bilbo attended the Council of Elrond where the history of the ring was explained by Gandalf and Elrond and the plan was made to destroy it in Mount Doom.
'That is the warning bell for the Council of Elrond,' cried Gandalf. 'Come along now! Both you and Bilbo are wanted.'
(source: The Fellowship of the Ring)
As @KevinTroy notes, Bilbo even volunteered before Frodo:
'Very well, very well, Master Elrond!' said Bilbo suddenly. 'Say no more! It is plain enough what you are pointing at. Bilbo the silly hobbit started this affair, and Bilbo had better finish it, or himself.
(source)
Tolkien Gateway states Bilbo's ageing is caused by the Ring's destruction:
Because of the Ring's destruction, that prolonged his life, Bilbo started ageing again; as Arwen told to Frodo later, Bilbo started looking ancient according to his kind.
It doesn't seem too far-fetched to me to infer that the apparent dementia is a likely result. It's based on the following quote:
‘Do you wonder at that, Ring-bearer?’ said Arwen. ‘For you know the power of that thing which is now destroyed; and all that was done by that power is now passing away. But your kinsman possessed this thing longer than you. He is ancient in years now, according to his kind; and he awaits you, for he will not again make any long journey save one.’
(source: The Return of the King)
As @chepner notes, Bilbo seems to recall some of his memories afterwards:
‘What a pity!’ said Bilbo. ‘I should have liked to see it again. But no, how silly of me! That’s what you went for, wasn’t it: to get rid of it? But it is all so confusing, for such a lot of other things seem to have got mixed up with it: Aragorn’s affairs, and the White Council, and Gondor, and the Horsemen, and Southrons, and oliphaunts – did you really see one, Sam? – and caves and towers and golden trees, and goodness knows what besides.
(source)
