Learn Hardcore French: Marie étudie à l'université, et la matière qu'elle aime le plus est la biologie. - Marie studies at the university, and the subject she likes the most is biology.

The noun université starts with a vowel sound, and the preposition + article here is à + la.
In French, à + la becomes à l' before a vowel sound:

à + la université → à l'université

This is purely a pronunciation/spelling rule; the meaning stays the same. You must write and say à l'université, never à la université.

Why do we use à l'université rather than something like en université or dans l'université?

For institutions (school, university, church, hospital, etc.), French normally uses à:

à l’université – at university
à l’école – at school
à l’hôpital – at the hospital

En and dans are used differently:

en is used with many countries and some abstract locations: en France, en classe.
dans tends to mean physically inside something concrete: dans l’université = inside the (specific) university building.

If you just mean “she studies at university (as an institution)”, you say à l’université.

In English we say “at university” (without “the”). Why does French say à l'université, with l'?

French almost always uses an article (le, la, l’, les) in front of nouns, even where English leaves it out.

So:

English: at university
French: à l’université (literally “at the university”)

You only drop the article in a few fixed expressions (e.g. newspaper headlines, labels, etc.), but for a normal sentence like this, you must say à l’université.

How is étudie formed, and what tense is it?

Étudie is the present tense, 3rd person singular, of the verb étudier (“to study”).

Étudier is a regular -er verb:

j’étudie
tu étudies
il / elle / on étudie
nous étudions
vous étudiez
ils / elles étudient

Since Marie is elle (she), you use the il/elle/on form: Marie étudie.

What’s the difference between matière and words like cours or “subject”/“class” in English?

Matière here means an academic subject/field, like:

la biologie – biology
les mathématiques / les maths – mathematics
la chimie – chemistry

Cours usually means a course/class (a specific class you take):

un cours de biologie – a biology course/class

So:

la matière qu’elle aime le plus ≈ “the (academic) subject she likes most”
le cours qu’elle aime le plus ≈ “the specific class she likes most”

Using matière in this sentence is natural because it’s about her favorite subject in general.

Why is it la matière qu’elle aime le plus and not la matière qui elle aime le plus?

This is about relative pronouns qui and que.

In la matière qu’elle aime le plus:

la matière is the thing being referred to again.
Inside the clause (elle aime …), la matière is the direct object of aimer (she likes it).

In French:

Use qui when the antecedent is the subject of the verb in the relative clause.
Use que (qu’) when the antecedent is the direct object.

Here, elle is the subject of aime, and “the subject/thing she likes” is the object, so we use que:

la matière qu’elle aime = the subject (that) she likes

Qui elle aime is wrong here; you must say qu’elle aime.

Why is que shortened to qu’ in qu’elle?

This is another case of elision.

Que ends with a vowel sound, and elle starts with a vowel sound. French avoids two vowel sounds meeting like that, so:

que + elle → qu’elle

The same happens with many small words ending in a vowel: je → j’, ne → n’, si → s’ (in s’il), etc., before words starting with a vowel sound.

Why do we say le plus and not la plus, since matière is feminine?

In la matière qu’elle aime le plus, le plus is an adverb (“the most”) modifying the verb aimer (“likes the most”).

When plus modifies a verb or adverb, you use le plus, and it does not agree with the noun:

Elle aime cette matière le plus. – She likes this subject the most.

You only use la plus / le plus / les plus to agree with a noun when plus is part of an adjective phrase:

C’est la matière la plus intéressante. – It’s the most interesting subject.

So:

la matière qu’elle aime le plus – the subject she likes the most (focus on the verb “likes”)
la matière la plus intéressante – the most interesting subject (focus on the adjective “interesting”)

Why are both la matière and la biologie feminine?

In French, every noun has a grammatical gender, masculine or feminine. You usually have to memorize it:

la matière – feminine
la biologie – feminine

Many nouns ending in -tion, -sion, -té, -ie are feminine, so biologie (ending in -ie) is feminine and takes la.

That’s why the sentence has:

la matière … est la biologie

Could we change the word order, like in English: “Biology is the subject she likes the most”?

Yes. A very natural French version is:

La biologie est la matière qu’elle aime le plus.

You can also say, with a slightly different emphasis:

La matière qu’elle aime le plus, c’est la biologie.

All of these are correct; they just put the focus on different parts of the sentence, as in English.