Babaannemle yengem mutfakta konuşurken ben de masaya tabakları dizdim.

-le means with, but in everyday Turkish it can also link two nouns in a way that feels like and.

So babaannemle yengem means:

• my grandmother and my sister-in-law/aunt-by-marriage

You could also say babaannem ve yengem. Both are correct, but -le is very common and natural in speech.

In Turkish, when -le/-la is used to join two nouns, it usually appears only on the first one:

• annemle babam = my mother and my father
• babaannemle yengem = my grandmother and my sister-in-law/aunt-by-marriage

So this is a normal pattern. You do not need to repeat it on both nouns.

That -m is the 1st person singular possessive ending, meaning my.

• babaanne = paternal grandmother

• babaannem = my paternal grandmother

• yenge = sister-in-law / aunt-by-marriage
• yengem = my sister-in-law / my aunt-by-marriage

So the sentence is talking about my relatives.

Yenge can mean different female in-law relationships depending on context. Very commonly it means:

• sister-in-law
• or a woman married into the family, sometimes understood as aunt-by-marriage

So the exact English word depends on the family relationship, but the Turkish word itself is very normal and common.

Konuşurken means while talking / while speaking.

It comes from konuşmak (to talk, to speak) plus -ken, which gives the idea of while or when.

So:

• konuşmak = to talk
• konuşurken = while talking

In this sentence, it sets the background action:

• while my grandmother and my sister-in-law were talking...

Turkish does not need a separate word for were here the way English does.

Not by itself. Konuşurken only gives the idea while talking.
The subject comes from the words before it:

• Babaannemle yengem ... konuşurken
  = while my grandmother and my sister-in-law were talking

So the people doing the talking are understood from the noun phrase, not from a special ending on konuşurken.

This is because of consonant harmony.

The locative ending is basically -da / -de / -ta / -te, meaning in / at / on.
After a voiceless consonant like k, Turkish uses -ta / -te.

• mutfak = kitchen
• mutfakta = in the kitchen

So mutfakta is the correct form.

Because this de means too / also, and that word is written separately in Turkish.

• ben de = I too / I also / as for me, too

This is different from the locative suffix -de/-da, which is attached to the word:

• mutfakta = in the kitchen
• evde = at home

So:

• ben de → separate, because it means also
• evde → attached, because it means in/at

Here ben de has the sense of I too, I also, or even as for me / meanwhile I.

It does not necessarily mean that the speaker was doing the same action as someone else. It simply adds the speaker as another participant in the situation.

So the feeling is something like:

• While they were talking, I, too / meanwhile I, arranged the plates on the table.

That use of de is very common in Turkish.

Because -ya / -ye here is the dative ending, which often shows direction toward something.

• masa = table
• masaya = to the table
• masada = on/at the table

With dizmek in this sentence, the idea is that the plates were being arranged onto / toward the table, not just located there.

So:

• masaya tabakları dizdim = I arranged/set the plates onto the table

That -ı is the accusative marker, used for a definite direct object.

• tabaklar = plates
• tabakları = the plates

So the speaker is talking about specific plates, not just plates in general.

Compare:

• tabak dizdim = I arranged plates
• tabakları dizdim = I arranged the plates

Yes, in some contexts tabakları can also mean his/her plates or their plates, because Turkish forms can be ambiguous.

But in this sentence, the natural reading is:

• tabakları = the plates (definite accusative plural)

The sentence structure makes that the most likely meaning.

Dizdim is the past tense, 1st person singular form of dizmek.

• dizmek = to line up, arrange, set out
• dizdim = I arranged / I lined up / I set out

In this context, it suggests placing the plates in order on the table. Depending on the translation, English might say:

• I arranged the plates
• I set out the plates
• I laid out the plates

Yes. That is one of the main jobs of -ken in konuşurken.

It means the action in the -ken clause was in progress while the main action happened:

• they were talking
• I arranged the plates

So the sentence gives a background action and a main action happening during that time.

Because Turkish normally places the main finite verb at the end.

So the structure is very natural:

• background clause: Babaannemle yengem mutfakta konuşurken
• main clause: ben de masaya tabakları dizdim

A very rough pattern is:

• While X was happening, I did Y.

Turkish word order is somewhat flexible, but having dizdim at the end is the standard, neutral pattern.