Gareth Rees has cited several 19th-century editors, some of whom saw two metaphors in these lines from Macbeth. Modern editors tend to agree to the extent that they see two plausible interpretations, but they differ in the details.
For example, G. K. Hunter writes in his edition (New Penguin Shakespeare. Penguin, 1967),
The horse imagery of 'Striding' and 'horsed' leads now (1) to a view of Macbeth's intention to murder as a horse that must be spurred, and (2) to a view of ambition (which could be a spur or stimulus) as a rider vaulting into his saddle, but overshooting the mark and falling on the other side.
A. R. Braunmuller's edition of Macbeth (New Cambridge Shakespeare. Cambridge University Press, 1997) also presents two interpretations of these lines. The first one is based on the observation that the lines continue the equine imagery from the preceding lines ("Striding the blast, or heaven's cherubin horsed ..."). Hunter points out a distinction that other editors don't mention:
Macbeth distinguishes his intent to murder, which he imagines as an unspurred horse, from his ambition to be king, which he imagines as an eager rider who overdoes his vault ("o'erleaps") and thus fails to land in the saddle; (...)
Braunmuller's second interpretation is that
horse and rider together fall when the pair fails to over-leap an obstacle.
For support for the second interpretation, Braunmuller refers to Catherine Belsey's paper "Shakespeare's 'vaulting ambition'" (English Language Notes, 1972), which associated Shakespeare's image with medieval and renaissance "depictions of Pride as a vaulting figure" (Braunmuller, page 133).
Both interpretations assume that Lady Macbeth's entrance prevents Macbeth from completing the sentence with the word "side".
Kenneth Muir's edition (The Arden Shakespeare. Routledge, 1962, 1984) adds, among other comments,
[John Dover] Wilson mentions that vaulting into one's saddle was a much-admired feat.
(John Dover Wilson's edition of Macbeth in the older Cambridge Shakespeare series was published in 1947.)
None of these editions follow Walter Savage Landor's suggestion to emend "itself" to "its sell" (i.e. its saddle).
Editors don't gloss "vaulting" because the verb vault here as the same meaning as today: "to jump or leap over". (For this reason, the verb has no entry in A Shakespeare Glossary by C. T. Onions, revised by Robert D. Eagleson; Oxford University Press, 1986.)
What the footnotes and endnotes cited above don't comment on is the concept of ambition, which is not listed in A Shakespeare Glossary, either. It can be instructive to look at other passages in Shakespeare's plays that use the words "ambition" or "ambitious". For example, Brutus says in Julius Caesar, Act 3, scene 2:
As Caesar loved me, I weep for him;
as he was fortunate, I rejoice at it; as he was
valiant, I honour him: but, as he was ambitious, I
slew him. There is tears for his love; joy for his
fortune; honour for his valour; and death for his
ambition.
Ambition is mentioned also in other passages in the play, for example, Antony's defence of Julius Caesar against Brutus's accusation of ambition.
Oliver says in As You Like It, Act 1, scene 1:
I'll tell thee, Charles:
it is the stubbornest young fellow of France, full
of ambition, an envious emulator of every man's
good parts, a secret and villanous contriver against
me his natural brother
In Henry IV Part I, Act 5, scene 4 Prince Henry speaks the words
Ill-weaved ambition, how much art thou shrunk!
These words are spoken after the death of Henry Percy (Hotspur), leader of the northern rebellion against King Henry IV.
These passages are intended to show that the word ambition in Shakespeare's works had stronger negative connotations than today; it is often associated with political figures who try to gain more power than was appropriate for their position in society. In Shakespeare's works, this type of ambition is not merely a matter between humans, but disturbs the natural order (see also the images of strange natural phenomena in Macbeth) and has more "cosmic" implications. For a fuller exposition of this connection between human and natural order, see Ulysses's speech on "degree" in Troilus and Cressida, Act 1, scene 3.
Finally, what the scholars cited in Gareth Rees's answer don't mention is that Macbeth messes up the order of the actions he is thinking about: you can only spur on a horse after mounting it, yet the spurring is what he mentions first. This is in keeping with his obsession with the future (prophecies, his succession, etcetera).