Babaannem de anneannem de çok misafirperver.

Turkish often makes finer family distinctions than English.

• babaanne = your father’s mother
• anneanne = your mother’s mother

So Turkish does not usually use just one general word in cases like this; it can specify which side of the family the grandmother is on.

In your sentence:

• babaannem = my paternal grandmother
• anneannem = my maternal grandmother

The -m is the 1st person singular possessive suffix, meaning my.

So:

• babaanne → babaannem = my paternal grandmother
• anneanne → anneannem = my maternal grandmother

Turkish often expresses possession with a suffix instead of a separate word like my.

Because Turkish usually builds possession directly into the noun.

Instead of saying something like my grandmother, Turkish says:

• anne = mother
• annem = my mother

The same pattern appears here:

• babaanne = paternal grandmother
• babaannem = my paternal grandmother

So the idea of my is already included in the word itself.

Here, de means also / too.

So the pattern:

• Babaannem de anneannem de ...

has the sense of:

• My paternal grandmother too, and my maternal grandmother too...
• more naturally in English: Both my paternal grandmother and my maternal grandmother...

It adds the idea that each one is included.

Because this de is a separate word, not an ending attached to the noun.

This is an important distinction in Turkish:

• de written separately = also / too
• -de written attached = in / on / at (the locative suffix)

For example:

• Annem de geldi. = My mother came too.
• Annemde kaldım. = I stayed at my mother’s place.

So in your sentence, de is separate because it means also/too.

This de/da follows vowel harmony.

• After front vowels (e, i, ö, ü) → de
• After back vowels (a, ı, o, u) → da

Both babaannem and anneannem end with the front vowel e in their final syllables, so de is used.

In Turkish, when you want to say both X and Y, a very common structure is:

• X de Y de ...

So:

• Babaannem de anneannem de çok misafirperver.

means that both grandmothers are hospitable.

The repetition is normal and natural. It gives a balanced meaning: both this one and that one.

Yes, you could say:

• Babaannem ve anneannem çok misafirperver.

That also means My paternal grandmother and my maternal grandmother are very hospitable.

The difference is mainly nuance:

• X ve Y = simple and
• X de Y de = both X and Y, often with a slightly stronger sense that each one equally fits the statement

So the original sentence is a bit more emphatic or balanced.

In Turkish, in the present tense, 3rd person, sentences with noun or adjective predicates often do not use an explicit word for is/are.

So:

• çok misafirperver = very hospitable

and the whole sentence can simply be:

• Babaannem de anneannem de çok misafirperver.

Literally, Turkish does not need to say are here.

A more formal or emphatic version could use -dir:

• Babaannem de anneannem de çok misafirperverdir.

But in everyday speech, leaving it out is completely normal.

Here, çok means very because it comes before an adjective.

• çok misafirperver = very hospitable

çok can mean different things depending on what it modifies:

• before a noun: many / a lot of
• before an adjective or adverb: very

So in this sentence, it is an intensifier: very.

Yes, misafirperver is an adjective meaning hospitable.

In Turkish, adjectives do not change for:

• singular/plural
• masculine/feminine
• person

So the same word works for one person or many people:

• Anneannem misafirperver.
• Babaannem de anneannem de misafirperver.

No extra agreement is needed on the adjective.

The sentence as given is very natural:

• Babaannem de anneannem de çok misafirperver.

Turkish word order is somewhat flexible, but different orders can change the emphasis.

This version is neutral and clear:

• first the people being talked about
• then the description

You may hear other orders in context, but for a learner, the given order is the safest and most natural one to use.

Turkish often drops things that are understood from context, but in this sentence the two nouns are important because they contrast both grandmothers.

If you replaced them with pronouns, you would lose that nice explicit distinction between the paternal and maternal grandmother.

So keeping both:

• babaannem
• anneannem

is exactly what makes the sentence precise and natural.