Mér finnst bókin sem þú lest mjög áhugaverð.

Because finnast uses the experiencer in the dative case.
So mér is the dative form of ég, and it means to me.

That is why Mér finnst ... is a very common Icelandic way to say things like:

• I think ...
• I feel ...
• I find ...

Literally, it is closer to To me, ... seems ... than to a direct English I ... sentence.

Not exactly.

It is related to finna, but in this construction it does not mean find in the sense of locate something.

Compare:

• Ég finn bókina. = I find the book / I locate the book
• Mér finnst bókin áhugaverð. = I find the book interesting / I think the book is interesting

So in your sentence, finnst is part of the opinion expression mér finnst.

Bókin means the book.

It contains:

• bók = book
• -in = the suffixed definite article, meaning the

So Icelandic often puts the definite article at the end of the noun instead of using a separate word like English the.

• bók = a book
• bókin = the book

Because bókin is the grammatical subject of finnst, so it is in the nominative case.

Even though the sentence begins with mér, that does not make mér the subject.
The structure is really more like:

The book seems very interesting to me.

So:

• mér = dative experiencer
• bókin = nominative subject

Sem introduces a relative clause. Here it means that or which:

• bókin sem þú lest = the book that you are reading

A useful thing for learners is that sem does not change form for gender, number, or case. It stays sem.

Here lest is the 2nd person singular present tense of lesa (to read).

In Icelandic, the simple present often covers what English expresses with either:

• you read
• you are reading

So in this sentence, sem þú lest naturally means that you are reading.

If you want to stress that the action is going on right now, Icelandic can also use:

• sem þú ert að lesa

But sem þú lest is perfectly normal.

Because Icelandic is generally a verb-second language in main clauses.

Here, the first element is mér, so the finite verb finnst comes next:

• Mér | finnst | bókin ...

That word order is very natural in Icelandic.

You could also put the subject first:

• Bókin sem þú lest finnst mér mjög áhugaverð.

That is also correct, but Mér finnst ... is extremely common.

Because adjectives in Icelandic agree with the noun they describe in gender, number, and case.

Here the noun is bókin, which is:

• feminine
• singular
• nominative

So the adjective must match, giving áhugaverð.

Compare:

• áhugaverður = masculine
• áhugaverð = feminine
• áhugavert = neuter

So the form is chosen to match bókin.

Because áhugaverð is not being used directly inside the noun phrase like an interesting book.
Instead, it is a predicate adjective after the verb finnst.

So the sentence is structured like this:

• bókin sem þú lest = the noun phrase
• mjög áhugaverð = what the speaker says about that noun phrase

In other words, it is like:

The book that you are reading is very interesting to me.

Not:

the very interesting book that you are reading

Usually no. In standard Icelandic, sem is normally kept in this kind of relative clause.

So:

• bókin sem þú lest = normal
• bókin þú lest = not normal standard Icelandic

English often allows that to disappear, but Icelandic usually does not here.