Η γιαγιά μου λέει ότι η περιποίηση των φυτών την χαλαρώνει.

Here μου means my.

Greek often uses a weak genitive pronoun after a family noun:

• η γιαγιά μου = my grandmother
• literally, something like the grandmother of me

So even though μου can also mean to me with a verb, in this sentence it is possessive and belongs with γιαγιά.

Greek normally keeps the definite article in phrases like this.

So:

• η γιαγιά μου = my grandmother
• not just γιαγιά μου in neutral standard usage

This is very common with family members, body parts, and many possessive noun phrases. Greek is more comfortable than English with saying literally the grandmother my.

λέει is the 3rd person singular present form of λέω (to say).

So:

• λέω = I say
• λες = you say
• λέει = he/she/it says

Since the subject is η γιαγιά μου (my grandmother), Greek uses λέει: my grandmother says.

Here ότι means that and introduces a clause:

• λέει ότι... = she says that...

It connects the main clause to what is being said.

A useful extra note:
ότι = that
but ό,τι = whatever / anything that

So in your sentence, it is the conjunction that.

περιποίηση is a noun meaning care, tending, or looking after.

So:

• η περιποίηση των φυτών = the care of the plants
• more naturally in English: taking care of the plants or plant care

Greek often uses a noun where English might prefer an -ing form.

Because των φυτών is a genitive phrase meaning of the plants.

• των = of the (genitive plural article)
• φυτών = genitive plural of φυτά (plants)

So:

• η περιποίηση των φυτών = the care of the plants

After a noun like περιποίηση (care), Greek commonly uses the genitive to show what the care is directed toward.

Because περιποίηση is also a feminine singular noun, so it takes the feminine singular definite article η.

So the sentence has two different feminine nouns, each with its own article:

• Η γιαγιά μου
• η περιποίηση των φυτών

That is completely normal in Greek.

την means her here, and it is the direct object of the verb.

• χαλαρώνει = relaxes
• την χαλαρώνει = relaxes her

So the structure is:

• η περιποίηση των φυτών = the subject
• την = her
• χαλαρώνει = relaxes

In other words: Taking care of the plants relaxes her.

Greek weak object pronouns usually go before the verb, which is why you get την χαλαρώνει, not χαλαρώνει την in the neutral order.

It refers back to η γιαγιά μου.

So the sentence means:

• My grandmother says that taking care of the plants relaxes her.

The pronoun is feminine singular, which matches γιαγιά.

Because την is accusative, and the verb χαλαρώνει takes a direct object here.

• την = her as a direct object
• της usually means her / to her / of her in the genitive

Compare:

• την χαλαρώνει = it relaxes her
• της μιλάει = he/she speaks to her

So in this sentence, την is the correct form.

Because the real subject is not plants. The subject is the whole phrase:

η περιποίηση των φυτών = the care of the plants

The head noun there is περιποίηση, which is singular.
So the verb is singular too:

• η περιποίηση ... χαλαρώνει

The plural φυτών is only part of the of the plants phrase.

Yes. Greek word order is fairly flexible.

The version you have is a natural, neutral order:

• Η γιαγιά μου λέει ότι η περιποίηση των φυτών την χαλαρώνει.

But Greek could move things around for emphasis, for example:

• Η γιαγιά μου λέει ότι την χαλαρώνει η περιποίηση των φυτών.

That still means the same thing, but it gives different emphasis.

One thing that usually stays the same is that the weak object pronoun την comes before the verb:

• την χαλαρώνει

Yes, often it could.

In many everyday sentences:

• λέει ότι...
• λέει πως...

both can mean says that...

In your sentence, ότι is perfectly natural and a little more textbook-neutral. A learner should understand both.