It is never spelled out in much detail what it is like for Merlyn to experience living backwards in time. The story begins with a few vague allusions to Merlyn knowing and understanding things that he would not be expected to, by virtue of having seen them before; but initially, it is not clear to the reader by what mechanism he has seen the future already. However, he soon explains his situation to Arthur in chapter 3 of The Sword in the Stone:
I was unfortunately born at the wrong end of time, and I have to live backwards from in front, while surrounded by a lot of people living forwards from behind.
This explains that the wizard's hazy knowledge of the future is really him remembering things that he has already seen.
However, the haziness of Merlyn's future knowledge is often an important factor. He has seen much of the future, but he does not do a particularly good job of recognizing when he has reached a crucial juncture, often arriving to impart his foresight only after the point at which it would be useful. (From Merlyn's viewpoint, this may mean that he actually speaks up too early, rather than too late, but I do not recall that oddity ever being dwelt upon.) His haziness is also thematically tied to his long sleep, into which he he knows he will be placed by Nimue's spell. He is always anticipating and dwelling upon his upcoming ensorcellment (the fallout of his failed romance with Nimue), and seems to be metaphysically groggy throughout much of the narrative.
Matters are further confused by White's explicit use of anachronisms to give the story a better atmosphere. Very early in The Sword in the Stone, Sir Ector is described as drinking port, which the narration admits is not really port, but describing it as port should give the reader the right idea; and it is stated that anachronistic substitutions will henceforth be used whenever it will be useful, based on what the reader is expected to be familiar with. (In fact, the whole Anglo-Norman setting of the story seems to be one such anachronism, since the High Middle Ages are a more familiar setting to most readers than sixth-century post-Roman Britain.) One further consequence of the story's heavy use of anachronistic elements is that when Merlyn mentions something ostensibly from the future, it is sometimes unclear to the reader whether what he is talking about is supposed to be familiar to Arthur or not.
All of these factors combine to make Merlyn a very confusing character, both for the other characters in the narrative (especially Arthur, who pays by far the most attention to Merlyn early on) and for the reader. This all seems to be intentional on White's part, and it is very hard to get a clear idea of what life is like for Merlyn himself; the wizard remains, appropriately, an enigma.
Incidentally, if you are a fan of White's Merlyn, then I would recommend reading the originally published 1938 standalone version of The Sword in the Stone, rather than the abridged version that White incorporated into 1958's The Once and Future King. Several episodes from the wizard's training of young Wart are cut in The Once and Future King, and I think it is better to read them at the points in the story where they originally appeared, although there is also another major episode that only appears in the 1958 version.