Learn French: Quelle est la taille de cette robe ?

Quelle agrees with the noun it is asking about, which is la taille (the size). Because taille is feminine, we use the feminine form Quelle. Notice that they still agree even when they are separated by the verb est.

The English translation is "What size is this dress?" Why is the French literally "What is the size of this dress?"

In French, you cannot say Quelle taille est cette robe ?. When asking for a measurement or size using the verb être (to be), you usually have to rephrase it to "What is the [noun] of..." (Quelle est la taille de...).

Can I use taille to ask for a shoe size too?

No, taille is only used for clothing sizes, a person's height, or the physical dimensions of an object. If you want to ask about shoe size, you must use a different word: la pointure.

Quick refresher: why do we use cette here instead of ce?

Cette is the feminine singular form for "this" or "that". Since robe (dress) is a feminine word, we must use cette. For masculine nouns, you would use ce (or cet before a vowel).