Anneannem, kayısı reçelinin şekerini azalttığını söyledi.

Because -m is the 1st person singular possessive suffix, so anneannem means my maternal grandmother.

• anneanne = maternal grandmother
• anneannem = my maternal grandmother

With family words, Turkish very often uses possessive endings when talking about your own relatives:

• annem = my mother
• babam = my father
• teyzem = my aunt
• anneannem = my maternal grandmother

So this is very natural Turkish.

The comma is not required by the core grammar of the sentence. It mainly marks a slight pause or gives Anneannem a topical, emphasized feel.

Many people would simply write:

Anneannem kayısı reçelinin şekerini azalttığını söyledi.

So the comma is stylistic here, not something you must always copy.

Kayısı reçeli is a very common Turkish noun compound pattern.

• kayısı = apricot
• reçel = jam
• kayısı reçeli = apricot jam

In this pattern, the first noun usually stays plain, and the second noun takes a compound ending, often -i / -ı / -u / -ü.

Other examples:

• elma suyu = apple juice
• domates çorbası = tomato soup
• köy ekmeği = village bread

So kayısı reçeli is just the normal way to say apricot jam.

Because it is the possessor of the next noun, şekeri.

Turkish often uses a genitive + possessed noun structure:

• kayısı reçelinin şekeri = the sugar of the apricot jam

Here is the breakdown:

• kayısı reçeli = apricot jam
• kayısı reçelinin = of the apricot jam

The ending -nin is the genitive case here.

Şekerini can be broken down like this:

• şeker = sugar
• şekeri = its sugar / the sugar of it
• şekerini = its sugar, as a definite direct object

More specifically:

• -i = 3rd person possessive
• -ni = accusative, with n as a buffer consonant after the possessive ending

So in kayısı reçelinin şekerini:

• kayısı reçelinin = of the apricot jam
• şekerini = its sugar

Together: the sugar of the apricot jam

The accusative is there because this noun is the direct object of azaltmak inside the reported clause.

Because after söyledi, Turkish usually uses a reported clause in a nominalized form.

So instead of saying something like:

• ... azalttı söyledi — not correct

Turkish says:

• ... azalttığını söyledi = said that he/she reduced it

This form turns the action into something like the fact that he/she reduced it, which can then become the object of söyledi.

A rough breakdown:

• azalt- = reduce
• -tığ-ı = a nominalized form meaning something like that he/she reduced
• -nı = accusative, because the whole clause is the object of söyledi

So azalttığını söyledi means said that he/she reduced.

Because azalmak and azaltmak are different:

• azalmak = to decrease, to become less
• azaltmak = to reduce something, to make something less

In this sentence, there is a direct object:

• şekerini = its sugar

So the meaning is to reduce the sugar, not the sugar decreased by itself.

That is why azaltmak is the correct verb here.

Not 100% from grammar alone. The subject of the reported clause is not stated explicitly, so Turkish leaves it to context.

Very often, listeners will understand it as the same 3rd-person subject as the main clause, so the natural interpretation is:

• My grandmother said that she reduced the sugar of the apricot jam

But if you want to make the subject explicit, Turkish can do that by putting it in the genitive:

• Ayşe'nin kayısı reçelinin şekerini azalttığını söyledi. = He/She said that Ayşe reduced the sugar of the apricot jam.

So the sentence as written usually suggests she said that she did it, but context is what really settles it.

Because Turkish normally prefers Subject–Object–Verb order, and the main verb often comes last.

Here the whole reported clause:

• kayısı reçelinin şekerini azalttığını

is the object of:

• söyledi

So the structure is basically:

• Anneannem = subject
• kayısı reçelinin şekerini azalttığını = what she said
• söyledi = said

That final-verb pattern is very typical in Turkish.

It is reported speech.

Turkish often reports speech with a nominalized clause plus a verb like söylemek.

So:

• ... azalttığını söyledi = said that he/she reduced it

If you wanted direct speech, you would give the exact words. For example, if the grandmother was speaking about herself:

• Anneannem, Kayısı reçelinin şekerini azalttım, dedi.

That is direct speech.
The original sentence is the indirect, reported version.

Yes. Turkish has several natural ways to report the same idea, depending on style and emphasis.

For example:

• Anneannem kayısı reçelinin şekerini azalttığını söyledi.
• Anneannem, kayısı reçelinin şekerini azalttığını söyledi.
• Anneannem kayısı reçelinin şekerini azalttığını söylemiş.
  This adds a hearsay or reported-information feel.

But the important grammar in your sentence is the same:

• possessive/genitive chain in kayısı reçelinin şekerini
• reported clause in azalttığını
• main reporting verb söyledi