As the post referenced below notes, this is a malapropism. Shakespeare was fond of giving comic characters this type of misspeaking, as it gratified the Elizabethan audience's love of wordplay. In Much Ado About Nothing, for instannce, this is a more constant feature of Dogberry's characterization.
At the same time, there is often a certain logic in the word confusion. If an "exposition of sleep" should read "disposition to sleep," we should make a comparison between the two words and try to find a deeper logic governing that switch. When he awakens and speaks of "Bottom's dream," the same type of word switching occurs. He essentially paraphrases St. Paul but confuses what body parts can do:
The eye of man hath not heard, the ear of man hath not seen, man’s hand is not able to taste, his tongue to conceive, nor his heart to report what my dream was.
Of course, Bottom has a tendency to overreach in his display of sophistication and knowledge. We see that when he wants to take over the management of the play as well as play all the parts. Discovering himself much admired by the fairies, he again puffs himself with language to match the regal setting of Titania's fairy world. A "disposition" to sleep would seem more refined to Bottom than a desire to sleep. In Bottom's mind, "disposition" and "exposition" are close enough that he falters and chooses the wrong word, not likely knowing the meaning of either.
An exposition is a setting or showing forth. To write an exposition is to write a story or an explanation that puts forth an understanding of the thing mentioned. When Bottom sleeps, and when the lovers sleep, much of the seemingly mad action of the night dissipates. Bottom returns to his human form and the lovers are sorted properly. Oberon and Titania make amends, and the Duke becomes willing to overrule Egeus in favor of the young lovers. The nightmare of the middle of the play sorts itself out through the dream-like quality of the play, and all is well.
The play itself is an "exposition" of sleep and of dreams and of "airy nothings" such as Theseus mentions:
More strange than true: I never may believe
These antique fables, nor these fairy toys.
Lovers and madmen have such seething brains,
Such shaping fantasies, that apprehend
More than cool reason ever comprehends.
The lunatic, the lover and the poet
Are of imagination all compact:
[. . .]
And as imagination bodies forth
The forms of things unknown, the poet's pen
Turns them to shapes and gives to airy nothing
A local habitation and a name.
Such tricks hath strong imagination,
That if it would but apprehend some joy,
It comprehends some bringer of that joy;
Or in the night, imagining some fear,
How easy is a bush supposed a bear!
Like the world of sleep and dreams, the world of theater functions to bring to the surface our latent anxieties and, when we are lucky, find a way to order our thoughts and desires such that we end in comedy rather than tragedy.
When Bottom says this to Titiana, he means that he is sleepy. However, he uses a malapropism when he states that he has an "exposition" for sleep. A malapropism occurs when a character gets confused and uses a wrong word that sounds close to the right word. Malapropisms are used for comic effect.
What Bottom means is that he has a "disposition," not an "exposition," for sleep. Using the wrong word adds more fuel to the comic quality of this character, who already does not know he has an ass's head and is asking for food that an ass would eat. Bottom now shows he is also an "ass," metaphorically speaking, for using the wrong word.
It is possible that Puck or Oberon have deliberately made both Bottom and Titania sleepy, for he and Oberon show up right after the twosome has fallen asleep. Having gotten the Indian boy, Oberon wants Puck to take away the spell that has made Titania fall in love with Bottom.