I understand (correct me if I'm wrong) that da is a short casual form of desu. But what does it mean in this phrase? or why it is used?
Tabeta bakari da.
[食]{た}べたばかりだ。(Tabeta bakari da)
I have just eaten.食べたばかりなの?(Tabeta bakari nano?)
Have you just eaten?食べたばかりではない。(Tabeta bakari dewa nai)
I have not just eaten.
There is a writing style named 論文調 that is for an essay in Japanese, and the end of sentence is 'da/dearu' meaning the assertion. It seems that any examples of the language textbooks use this style usually. That is a mechanical style at the opposite end of the polite style of expressions such a honorific language, and it is a kind of literary style.
ばかり【bakari】 is a 副助詞 (adverbial particle), which is derived from the 連用形 (-masu stem) of the verb はかる. But the particle (and 連用形 in general) behaves much like a noun. (Join to other noun-like words with の, make into a predicate by adding だ, etc.)
Now you essentially have a noun phrase 食べたばかり. To make a sentence out of this, you have to add だ・です (or だった・でした in the past tense).
This is the same as in
本だ。
hon da.
[This] is [a] book.
As you said, “だ” is a colloquial form of “です,” a predicate meaning “is, am,” and "食べたばかりだ” means “I’ve finished meal just now.”
“だ” here functions as I am in the state of having finished meal just now.
"da" is a short casual form of desu
. What exactly is your question?