Die Aubergine und die Süßkartoffel backen im Ofen, bis alles weich ist.

Because both nouns are feminine singular nouns in the nominative case here.

• die Aubergine = the eggplant
• die Süßkartoffel = the sweet potato

German nouns have grammatical gender, and both of these happen to be feminine, so their nominative singular article is die.

Because the subject is two things joined by und:

• die Aubergine und die Süßkartoffel

That makes the subject plural, so the verb must also be plural.

• die Aubergine backt = the eggplant bakes
• die Aubergine und die Süßkartoffel backen = the eggplant and the sweet potato bake

So here backen is the 3rd person plural form, not the infinitive.

Grammatically, it is a statement in the present tense:
The eggplant and the sweet potato bake in the oven...

In recipe language, German often uses:

• the imperative
• or the infinitive

So in a recipe, you might also see something like:

• Die Aubergine und die Süßkartoffel im Ofen backen, bis alles weich ist.
• Backe die Aubergine und die Süßkartoffel im Ofen, bis alles weich ist.

But the sentence you have is still understandable and natural enough.

im is a contraction of in dem.

• in dem Ofen → im Ofen

Here it means in the oven.

Because this is about location rather than movement, German uses the dative case after in:

• in dem Ofen / im Ofen = in the oven

Because bis introduces a subordinate clause, and in German, subordinate clauses are normally separated by a comma.

So:

• Die Aubergine und die Süßkartoffel backen im Ofen,
• bis alles weich ist.

That comma is required in standard German.

Because bis is a subordinating conjunction, and in German subordinate clauses, the conjugated verb goes to the end.

So the pattern is:

• bis
  • subject + other elements + verb

Here:

• alles = subject
• weich = adjective
• ist = verb at the end

So:

• bis alles weich ist

This is a very important German word-order rule.

alles means everything or all of it, referring to the ingredients as a whole.

So bis alles weich ist means something like:

• until everything is soft
• until it is all soft

If you wanted to focus specifically on the two items, you could also say:

• bis beide weich sind = until both are soft

Both are possible, but alles is very common when talking about food collectively.

Because alles is grammatically singular in German.

Even though it refers to more than one thing in meaning, the pronoun alles takes a singular verb:

• alles ist
• not alles sind

Compare:

• beide sind weich = both are soft
• alles ist weich = everything is soft

Because weich is a predicate adjective here, used after sein.

In German, adjectives after verbs like sein, werden, and bleiben are not inflected:

• alles ist weich
• die Aubergine ist weich
• die Kartoffeln sind weich

But adjectives before a noun do change:

• die weiche Aubergine
• die weiche Süßkartoffel

So weich stays unchanged because it comes after ist.

Because they are nouns, and all nouns are capitalized in German.

So:

• die Aubergine
• die Süßkartoffel
• der Ofen

This is one of the most noticeable spelling differences between German and English.

The ß is called Eszett or scharfes S.

In Süßkartoffel, it comes from süß = sweet.

So:

• Süßkartoffel literally means sweet potato

A useful spelling point:

• after a long vowel or diphthong, German often uses ß
• after a short vowel, it usually uses ss

Also, in Switzerland, ß is normally written as ss, so you may also see Süsskartoffel there.