Liomsa is an emphatic form of liom — using it in this sentence is the equivalent of “The book is mine.”
As with the English version, this sentence emphasises ownership:
"The book (which we previously mentioned) is mine."
Just out of interest, is this the same construction as 's ann leam in Scots Gaelic?
I think this is the same construction as 'Is leamsa ...' ('... is mine') in Scottish Gaelic. "'S ann leamsa a tha ..." would be equivalent to " ... belongs to me", which is subtly different in English, but I'm not sure there's a distinction between the two constructions in Gaelic.
"'S ann" is normlly use in Gàidhlig na h-Alba for fronting to indicate emphasis, "'S'ann leamsa a tha an leabhar" means roughly "It really is true that the book belongs to me", it's pulling "leamsa" forward to be in front of the verb (tha) so as to put put the emphasis on the distinction between possession (the "aig" in "agamsa") and ownership (the "le" in "leamsa"). But don't forget that "tha e leamsa" is rather a weaker statement than "is leamsa e" because the former expresses an incidental (and hence casually changeable) relationship with the book while the latter expresses an inherent (and hence requiring definite action if it is ever to be changed) relationship. I haven't managed to work out how this works in modern Gaeilge (but when reading older poetry - 17th and18th century Irish is vastly easier for me than modern Irish - I got the impresion that three or four centuries ago it was as it now is in Gàidhlig)
There's no verb in liom an leabhar. You need a verb for the sentence to make sense. The copula fills that role in this sentence.
"Liom" translates to "With me", hence "Is maith liom x" (x is good with me).
It's my book is more specifically Is é mo leabhar é, but context is important. If you were saying "It's my book, I've been looking for that everywhere", you'd say Is é mo leabhar é, bhí mé á lorg ar fud na háite. On the other hand the answer to "Whose book is this?" - Cé leis an leabhair seo? is Is liomsa an leabhar, which can reasonably interpreted as "the book is mine" or "it's my book", and you could even interpret the question and answer as "Who owns this book?" and "I own the book" ("own" can be a verb in English, but there is no equivalent verb in Irish).
So you're not wrong, but the phrase "it's my book" is ambiguous in English, and can mean two different things.
Semi-Related question. Is there a way to say "That is mine". Would it be "Is Liomsa Sin"? Or would that be wrong for a reason I've missed?
Is liomsa é sin or sin mo cheannsa (more completely is é sin mo cheannsa)