Мне нравятся книги, у которых крепкий переплёт.

The verb нравиться works differently from English to like.

With нравиться, the thing that is liked is the grammatical subject, and the person who likes it goes in the dative case.

So:

• мне = to me
• книги нравятся = books are pleasing

Literally, Мне нравятся книги is close to Books are pleasing to me.

Compare:

• Я люблю книги = I love books / I like books very much
• Мне нравятся книги = I like books

And я нравлюсь книгам would mean something strange like the books like me.

Because нравиться takes the experiencer in the dative.

Common pattern:

• Мне нравится... = I like...
• Тебе нравится... = You like...
• Ему нравится... = He likes...
• Нам нравятся... = We like...

So мне is not optional here; it is the normal form required by the verb нравиться.

Because the verb agrees with книги, which is plural.

• книга = singular
  
  • Мне нравится книга. = I like the book.
• книги = plural
  
  • Мне нравятся книги. = I like books.

Since книги is the grammatical subject, the verb must be plural: нравятся.

Книги is in the nominative plural.

That may feel surprising to an English speaker, because in English books looks like the object of like. But with нравиться, Russian treats the liked thing as the subject, not the object.

So in:

• Мне нравятся книги

the structure is basically:

• To me
  • are pleasing
    • books

That is why книги is nominative.

Because у которых крепкий переплёт is a relative clause describing книги.

Russian normally separates this kind of clause with a comma, just like English often does with clauses beginning with which, that, or whose.

So:

• Мне нравятся книги, у которых крепкий переплёт.

means:

• I like books that have a sturdy binding.

The comma is standard here.

Because у которых expresses possession: whose / which have.

The phrase у кого / у чего literally means something like at whom / at what, but very often in Russian it is used to mean someone has or something has.

So:

• книги, у которых крепкий переплёт

literally means:

• books, at which there is a sturdy binding

Natural English:

• books that have a sturdy binding
• books whose binding is sturdy

If you said книги, которые крепкий переплёт, that would be ungrammatical.

Because the preposition у requires the genitive case.

The base form is:

• которые = nominative plural

But after у, it changes to:

• у которых = of/at which ones

Since it refers back to книги (plural), the form is plural genitive: которых.

Yes, English speakers often feel that way.

Russian often omits есть (there is / have) in the present tense.

So:

• у которых крепкий переплёт

is understood as:

• у которых есть крепкий переплёт

Literally:

• which have a sturdy binding

Russian very often leaves out есть in present-tense possession statements.

Compare:

• У меня есть книга. = I have a book.
• У меня книга. = less standard in many contexts, but in some contexts the meaning is still understood

In your sentence, omission of есть is completely natural.

Крепкий переплёт is in the nominative singular.

That is because переплёт is the thing being said to exist in the relative clause:

• у которых (есть) крепкий переплёт

So переплёт acts like the subject of the implied есть.

Forms:

• крепкий = masculine nominative singular
• переплёт = masculine nominative singular

Not exactly.

• переплёт = binding
  This refers to the way the book is bound together, often suggesting the spine and the sturdiness of the physical construction.
• обложка = cover
  This is the outer cover of the book.

So крепкий переплёт means something like:

• sturdy binding
• solid binding
• strong binding

It is more specific than just saying the book has a nice cover.

Yes. A very natural simpler version is:

• Мне нравятся книги с крепким переплётом.

This means:

• I like books with a sturdy binding.

That version is shorter and probably more common in everyday speech.

Your original sentence is also correct, but it is a bit more explicit because it uses a relative clause:

• books that have a sturdy binding

instead of just:

• books with a sturdy binding

Not always. Крепкий has a range of meanings depending on context.

With переплёт, it means something like:

• sturdy
• solid
• strong
• durable

In other contexts, крепкий can also mean:

• strong tea or coffee
• strong alcohol
• a solid handshake
• a strong/healthy person

So here, крепкий переплёт is best understood as a sturdy or solid binding.

Usually it means books in general.

• Мне нравятся книги... = I like books...

Russian often uses the plural without an article for general statements, just as English can.

So the sentence describes a general preference: the speaker likes books that have sturdy bindings.

If you wanted to talk about specific books, the context would usually make that clear.