The Hobbit was not an adaptation, but a translation.
Tolkien envisioned himself as a translator of ancient manuscripts that had made their way into his hands. The Red Book of Westmarch, in which Frodo had compiled Bilbo's notes and consisted of There and Back Again and The Downfall of the Lord of the Rings and the Return of the King, amongst other works. Tolkien had acquired these works and translated them into Old English, and then further into Modern English (as outlined in this comment).
While never explicitly stated by the Professor himself, it is implicit in the way many of the sections of the Lord of the Rings (LR) outside the story were written. One example is the Prologue, written in the style of a modern editor describing a time long past. Other examples exist in an introductory note to the revised edition of the Hobbit (TH), the Preface to The Adventures of Tom Bombadil, and Appendixes A, D and F, especially the note on Shire records, which invents a manuscript tradition (that the original Red Book had not survived but a copy had).
As for the stylistic difference, this from an out-of-universe perspective is because of the later addition of TH to the Legendarium when Allan & Unwin had asked for a sequel (LR), and it is worth noting that Book 1 (the first half of the Fellowship of the Ring) is stylistically closer to TH than the rest of LR, which Tolkien claims to be intentional. In-universe according to the translator theory, the stylistic change would not have been because of an adaptation but because TH was penned by Bilbo Baggins, while LR was written by Frodo, after the War of the Ring. Book 1, could therefore be stylistically closer to TH because Frodo would've been able to tell Bilbo about his adventure thus far up to their meeting in Rivendell:
What about helping me with my book, and making a start on the next?
Then Bilbo would read passages from his book (which still seemed very incomplete). or scraps of his verses, or would take notes of Frodo's adventures.
Fellowship of the Ring - Book II, Chapter 2: Council of Elrond
Given that Bilbo had most of his own tale to write about during the War of the Ring, as well as the possibility of beginning on LR, this would explain the similarity in style between Book 1 and TH, and the sudden change in the rest of LR.
Below is a plot of the Transmission of the Legends in The Chronicles of Arda by Måns Björkman
Key:
" " = Important Work, * * = Author
( ) = Translations, _ _ = Regions
> = Direction of flow
*Quennar i Onótimo*
"Of the Beginning of Time..."
"Yénonótië"
"The Tale of Years"
*Rúmil* | "Parma Culuina"
"Annals of Aman"---<---| _Doriath_ |
"Ambarcanta" | "The Grey Annals" |
"Ainulindalë" | | |
| | | *Pengolodh* | *Dírhaval*
|--------->-----+----->----"Quenta Silmarillion"--<--"Narn i Chîn Húrin"
| "Lammas"
| |
|----------------<-------------------+
|
|------------>--------+--------->--------+
| | |
| _Númenor_ _Rivendell_
| "Indis i·Ciryamo" "Books of Lore"
| | |
| | |
| _Arnor and Gondor_ |
| | | *Bilbo Baggins*
| "Book of the Kings" | "My Diary"
| "Book of the Stewards" +-("Translations from the Elvish")
| "Akallabêth" |
"Quentalë | *Torhir Ifant* | *Frodo Baggins* |
Ardanómion" | "Dorgannas Iaur" | *Sam Gamgee* |
| | | |----->----"The Lord of the Rings" |
| | | | | |
| | | | "The Red Book of Westmarch"
| | | | *Findegil* |
| | | |-->--"The Thain's Book"------<------|
| | | | | [Many copies] _The Shire_
| *Ælfwine* | | | | "The Tale of Years"
("Quenta Silmarillion") | +----------->----------| |
("Annals of Aman") | | |
("Grey Annals") | | |
| | | |
+-------->------------|-------------------<----------------+-------<-------+
|
*J. R. R. Tolkien*
("The Hobbit")
("The Lord of the Rings")
("The Silmarillion")
