Learn Turkish: Kirli elbiseleri sabunla temizliyorum. - I am cleaning the dirty clothes with soap.

The word for "soap" is sabun. To say "with soap" or "by means of soap", we attach the suffix form of ile (with). Since sabun ends in a consonant and its last vowel is an undotted "u", A-type vowel harmony dictates we use the suffix -la.

The verb for "to clean" is temizlemek. Why is it spelled temizliyorum instead of temizleyorum?

This is due to vowel narrowing. When a verb stem ends in "e" or "a" (like temizle-), and you add the present continuous suffix -iyor, that final vowel gets "squeezed" or narrowed. Here, the "e" drops and is replaced by the "i" from -iyor, giving us temizliyorum.

Why is there an extra -i at the very end of elbiseleri?

That final -i is the specific direct object (accusative) suffix. The word is built from elbise (clothing) + -ler (plural). Because you are cleaning specific dirty clothes ("the dirty clothes" rather than just "clothes" in general), Turkish requires this suffix.

I learned that elbise means "dress". Why does it translate to "clothes" here?