Моя внучка умеет кое-как читать по слогам, но ей уже нравится открывать книги самой.

Because внучка is a feminine singular noun in the nominative case.

Russian possessive words like мой / моя / моё / мои have to agree with the noun in:

• gender
• number
• case

So:

• мой внук = my grandson
• моя внучка = my granddaughter

Here моя is the correct feminine singular nominative form.

Умеет comes from уметь, which means to know how to do something / to have the skill to do it.

That is why it fits well with читать. Reading is a learned skill.

• умеет читать = knows how to read
• может читать = can read, is able to read, is in a position to read

A useful contrast:

• Она умеет читать. = She knows how to read.
• Она не может читать без очков. = She can’t read without glasses.

So in this sentence, умеет focuses on the granddaughter’s level of reading ability.

Кое-как is an adverb meaning something like:

• barely
• so-so
• in a clumsy or makeshift way
• not very well

In this sentence it suggests that her reading is still weak and awkward, not smooth or confident.

So умеет кое-как читать means she can read, but only poorly / with difficulty.

As for spelling: кое-как is simply written that way in Russian. Many words with кое- are hyphenated. It is a fixed spelling pattern you just have to learn.

Literally, по слогам means by syllables or syllable by syllable.

So читать по слогам means reading slowly by breaking words into syllables, the way a beginner reader often does.

• слог = syllable
• по слогам = syllable by syllable

This is a very common expression for early reading.

After по in this kind of meaning, Russian often uses the dative case.

Here слогам is the dative plural of слог.

This pattern often has a distributive or step-by-step sense:

• по одному = one by one
• по буквам = letter by letter
• по слогам = syllable by syllable

So по слогам is a standard expression built with по + dative.

Because the verb нравиться works differently from English to like.

Russian structures it more like to be pleasing to someone.

So:

• ей нравится literally means it is pleasing to her
• natural English meaning: she likes

That is why the person who experiences the liking goes in the dative case:

• она = she
• ей = to her

So ей уже нравится... means she already likes...

Because the grammatical subject of нравится is not книги.

What she likes here is the whole action:

открывать книги самой

That entire infinitive phrase functions as the subject:

• Ей нравится читать.
• Ей нравится гулять.
• Ей нравится открывать книги самой.

With an infinitive phrase like that, Russian uses нравится in the singular.

If the subject were actually books, then you would get plural:

• Ей нравятся книги. = She likes books.

But here it is not the books themselves; it is the activity of opening them herself.

Because the sentence is talking about general ability and general enjoyment of an activity, not one finished action.

• умеет читать = knows how to read
• нравится открывать книги = likes opening books / likes to open books

Russian normally uses the imperfective in these situations:

1. ability or skill
   • уметь читать
2. habitual or repeated actions
   • любить читать
   • нравится открывать

A perfective form would suggest a single completed act, which is not the point here.

Книги is the accusative plural of книга, because it is the direct object of открывать.

• книга = book
• книги = books

For inanimate nouns in Russian, the accusative plural is usually the same as the nominative plural. That is why the form looks unchanged.

Compare:

• Я открываю книги. = I open books.
• Книги лежат на столе. = The books are lying on the table.

Same form, different function.

Here самой means herself / on her own / by herself.

It refers back to the granddaughter, but because the structure is:

ей нравится + infinitive

the implied doer of открывать is linked to ей, which is dative. So Russian commonly uses the matching feminine singular form самой here.

So:

• ей нравится открывать книги самой = she likes opening books by herself

If the sentence were built with a nominative subject instead, you would expect сама:

• Она любит открывать книги сама.

So the difference is caused by the grammar of нравиться and the infinitive construction.

Уже means already or by now.

In this sentence it suggests a development or change of stage:

• she can only read with difficulty,
• but already she likes opening books on her own.

So уже gives the sentence a sense of progress: even though her reading is still limited, she has reached the point where books attract her independently.

This is a nuance point.

Since the granddaughter can only read кое-как and по слогам, the sentence does not present her as a real fluent reader yet. So открывать книги самой emphasizes that she likes to:

• take books herself,
• open them herself,
• interact with them independently,
• begin exploring them on her own

It sounds very natural when talking about a small child: she may not truly be reading well yet, but she is already drawn to books and wants to handle them herself.