Learn Hardcore French: À la bibliothèque, le silence aide Marie à lire son roman. - In the library, silence helps Marie read her novel.

À la bibliothèque → At the library
If you wanted to insist more on inside the building, you could also say Dans la bibliothèque (In the library), but À la bibliothèque is the most natural way to say at the library as a general location.
You would not say au bibliothèque, because bibliothèque is feminine, so it must be à la, not au (which is à + le for masculine nouns).

Why is there a comma after À la bibliothèque?

The comma separates an introductory location phrase from the main clause.

À la bibliothèque = Where? (setting)
le silence aide Marie à lire son roman = What happens?

French often puts a place or time expression at the beginning of the sentence, followed by a comma:

Aujourd’hui, je travaille. – Today, I am working.
Le soir, il lit. – In the evening, he reads.

You could also write the sentence without moving the phrase:
Le silence, à la bibliothèque, aide Marie à lire son roman. (less natural in everyday speech)
or
Le silence à la bibliothèque aide Marie à lire son roman. (no comma, slightly different rhythm).

But the given version, with the place phrase first and a comma, is very natural and neutral.

Why do we say le silence with le? Why not just silence without an article?

In French, you almost always need an article (or another determiner) before a singular noun.

le silence = the silence
un silence = a silence
du silence = (some) silence (partitive sense, less common here)

Here, le silence refers to silence in general in that place – almost like a concept: the silence (that exists there) helps her to read.

In English, you might naturally say Silence helps Marie read her novel (no the), but in French you must choose an article, and le (the definite article) is the most natural choice here.

What form of the verb is aide, and why is it spelled that way?

Aide is the third person singular present tense of the verb aider (to help).

Conjugation of aider in the present tense:

j’aide – I help
tu aides – you help (singular, informal)
il / elle / on aide – he / she / one helps
nous aidons – we help
vous aidez – you help (plural or formal)
ils / elles aident – they help

In the sentence:

Subject = le silence
Verb = aide

So we use aide (third person singular) because le silence is singular.

Why is it aide Marie à lire and not aide Marie lire or aide Marie de lire?

With aider, the usual structure in French is:

aider + quelqu’un + à + infinitive
→ to help someone to do something

So:

aider Marie à lire = to help Marie to read
aider les enfants à comprendre = to help the children understand

Forms like:

aider Marie lire – incorrect in standard French
aider Marie de lire – incorrect here (French does not use de after aider in this structure)

So the double à you see (à la bibliothèque … aide Marie à lire) is normal:
first à for the location (à la bibliothèque), and second à introducing the infinitive (à lire).

Can the word order be Le silence à la bibliothèque aide Marie à lire son roman? Is that still correct?

Yes, it is grammatically correct:

Le silence à la bibliothèque aide Marie à lire son roman.

The meaning is almost the same, but the emphasis is slightly different:

À la bibliothèque, le silence aide Marie à lire son roman.
Emphasis first on the place (we are talking about what happens at the library).
Le silence à la bibliothèque aide Marie à lire son roman.
Emphasis first on le silence; you then specify which silence: the one at the library.

Both are fine; the original is a very natural, neutral way to start from the context (the library).

Why is it son roman and not sa roman? I thought Marie is feminine.

In French, son / sa / ses agree with the gender and number of the noun possessed, not with the person who owns it.

son = his / her / its (before a masculine singular noun)
sa = his / her / its (before a feminine singular noun)
ses = his / her / its (before a plural noun)

Here:

roman (novel) is masculine singular, so you must use son: son roman.
It does not matter that Marie is female; the agreement is with roman, not with Marie.

Examples:

Marie lit son roman. – Marie is reading her novel.
Marie lit sa revue. – Marie is reading her magazine (revue is feminine).
Marie lit ses livres. – Marie is reading her books (plural).

Context tells you whether son / sa / ses mean his, her, or its.

What exactly is the difference between roman and livre?

Both words exist in French, but they are not the same:

un roman = a novel (a long fictional story)
un livre = a book (any kind of book: novel, textbook, cookbook, etc.)

So son roman is more specific: it tells us that Marie is reading her novel, not just any kind of book.

You could say son livre, but then you only say she is reading her book, without specifying that it is a novel.

How do I know the genders: la bibliothèque, le silence, le roman?

You mainly have to memorize noun genders in French, but there are some patterns that help:

la bibliothèque – feminine
Many words ending in -thèque (like bibliothèque) are feminine.
le silence – masculine
Many words ending in -ence / -ance are feminine (la différence, la chance),
but le silence is an exception: it is masculine and must be learned as le silence.
le roman – masculine
Many nouns ending in -an are masculine (le restaurant, le volcan, le roman).

When you learn a new noun, it is good practice to learn it with its article:

la bibliothèque (feminine)
le silence (masculine)
le roman (masculine)

Could we replace Marie with a pronoun? For example, say Le silence l’aide à lire son roman?

Yes, if it is already clear from context who you are talking about, you can replace Marie with the object pronoun la or le, which becomes l’ before a vowel sound:

Le silence aide Marie à lire son roman.
Le silence l’aide à lire son roman. – The silence helps her to read her novel.

Here, l’ stands for la (her), because aide begins with a vowel sound, so la aide → l’aide.

The meaning is the same; the version with Marie simply names her explicitly.