This is a complex question, for reasons I'll get into further down, but here's the Truth, as best as I've been able to determine based on pre-fourth edition sources:
Nobody knows. The D&D Multiverse is so ancient, so full of history, that its origins are lost even to myth. Here's my evidence:
As has been established over the years, even the gods of most D&D settings are not immortal, as you can see their corpses floating in the Astral Plane, and there's at least one account of a new god undergoing apotheosis. And particular prime worlds can be older or younger than others: The Prime World of Arthas, for instance, is supposedly so ancient that all its original gods have died, and other worlds have histories with wildly varying lengths; While the tendency for timelines to get fuzzy at the mythological end makes comparison less than exact, it seems reasonable to conclude they weren't all created at the same time. Even the Planes themselves aren't eternal and unchanging: Some rare adventures have contained descriptions of entire planes drifting away from the great wheel, or running aground, merging, splitting, and being created and destroyed - and on a more immediate level, there was at least one adventure where adventurers could play a key role in causing a plane to gain or lose land from its infinite expanse. I think it's reasonable, therefore, to conclude that the multiverse of D&D is not static, but gradually changing; That its gods, races and even planes die and are replaced by newcomers just as mortal NPCs are, albeit on a much grander scale. What this means is that it's rather hard to 'carbon date' the multiverse, as there's no part of it we can guarantee was around when it started.
Moreover, there is substantial evidence that the multiverse is very, very old: For example, there are situations that would have been huge when first introduced whose origins have been forgotten entirely: The Blood War has been waged for so long that there's no record of its origin even in myth and there are agreements that bind all the deities, even ones who refuse to get along with each other in any other respect, that have stood unopposed for so long that there is no mortal record of their ever being broken. If even the origins of such ancient and hard-to-forget things is lost, what hope has original creation? (Admittedly, there are a few obscure creatures and that could, potentially, have been created before or at the same time as the known multiverse: The Lady of Blades, the Serpent, perhaps the Dark Powers of Ravenloft and a few obscure monsters; However, if they do know, they aren't telling.)
To sum up, while we might not know the origin of the multiverse as a whole, there's evidence that it's been around practically forever. (It's even possible that it's literally existed forever, in a "turtles all the way down" sense) Ultimately, the origin of the multiverse is less likely to matter than the origins of specific parts of it, which GMs are free to make up as they please.
Disclaimer: I've based this answer primarily on information from the Planescape campaign setting, since it is sort of 'outside' all the others - and given that that setting was designed with the intent of having players debate and question the nature of the world, providing a canonical explanation of how and why everything was created would be working against designer intent. I strongly suspect we'll never get a batter answer, however, as it is in the interests of every designer yet to come to keep the origin vague so as to allow GMs and players to create their own origin stories as appropriate to their campaigns. Very Planescape-appropriate, that: There need not be only one truth.
