De is used in French as the equivalent of English possessive -s. So think of it like 'the apple's juice'. Not really 'the juice of the apple'. If this doesn't work, another helpful idea is that A HEAP of sound and word choices in French are purely to sound nice. So it's just easier to say de instead of de la.
This is a difference between English and French: in English, nouns can be used as adjectives and in French, adjectives can become nouns (with the addition of a determiner).
It happens that "jus de pomme" is a "noun of noun", where "pomme" is a noun complement, whereas "apple juice" is a "noun phrase", where "apple" is used as an adjective.
From a previous comment:
That is because a literal translation to English leads to juice of apple which could easily lead an English speaker to wonder if the juice belongs to something or someone called apple.
English speakers abandon the French ..noun of noun = de/of ....rule. They take the of noun, drop the of and place the supporting noun in front of the main noun making the transposed noun look more like the adjective that it is.
Noun of noun requires de (jus de pomme) becomes ...drop the de, move pomme in front of jus and end up with apple juice.
French requires noun of noun = de. English almost always avoids it because it is confusing when rendered that way.
To expand slightly on the comment
noun of noun = of/de ......rule
noun/jus ...of/de...... noun/pomme
jus de pomme.
It is a rule. noun of noun = of/de
First of all, it wouldn't be "au", since "pomme" is a singular feminine noun, it would be "à la".
As for why we can't use it, that's because when we talk about food, we usually use "de xxx" when the final product is extracted from xxx (i.e. "un fromage de chèvre") and we use "au xxx" when xxx is an ingredient in the final product (i.e "un sandwich au fromage (de chèvre)"). This "rule" may have exceptions I'm not aware of.
"jus à la pomme" would mean a juice using apple, but would imply that the juice has other ingredients as well, and usually we would complete the sentence with "... et à la banane.". We can also use a construction like this : "un jus pomme-banane" or "un sandwich tomate-mozzarella". I'm not sure the hyphen is mandatory.
Yes, it is, because that is not the way that concept is expressed in English. You can sometimes use the phrase "apple's juice" ("She took a bite, and the apple's juice ran down her chin"), but unless we are talking about the juice of one particular apple, it is apple juice, not apple's juice or juice of apple.