Ŝi skribas la nomon de la filmo.

Because nomon is the direct object of the verb skribas.

In Esperanto, the direct object usually takes the ending -n.
So:

• la nomo = the name
• la nomon = the name (as the thing being written)

In this sentence, she is writing the name, so nomo becomes nomon.

Because filmo is not the direct object here.

The direct object is la nomon — that is the thing being written.
de la filmo is a prepositional phrase meaning of the film.

After the preposition de, you normally do not add -n just because of the preposition. So:

• de la filmo = of the film
• not de la filmon

Here de means of.

So la nomon de la filmo literally means:

• the name of the film

It shows a relationship between two nouns. This is a very common Esperanto pattern:

• la pordo de la domo = the door of the house
• la koloro de la aŭto = the color of the car

Because both nouns are definite in this sentence:

• la nomon = the name
• la filmo = the film

Esperanto uses la wherever English would naturally use the.
So if both parts are definite, both can take la.

Compare:

• la nomo de filmo = the name of a film
• la nomo de la filmo = the name of the film

Skribas is present tense.

Esperanto verb endings are very regular:

• -as = present
• -is = past
• -os = future
• -us = conditional
• -u = command / jussive
• -i = infinitive

So:

• ŝi skribas = she writes / she is writing
• ŝi skribis = she wrote
• ŝi skribos = she will write

It can mean either one, depending on context.

Esperanto does not usually make the same automatic distinction English does between:

• she writes
• she is writing

Both can often be expressed by ŝi skribas.

If the context makes it clear that the action is happening right now, ŝi skribas can mean she is writing. In a more general sense, it can mean she writes.

Yes, Esperanto word order is fairly flexible, but the most neutral order is:

• Ŝi skribas la nomon de la filmo.

Because la nomon has -n, you can often move it without losing the basic meaning. For example:

• La nomon de la filmo ŝi skribas.

This is still understandable, but it sounds more marked or focused, something like:

• The name of the film, she is writing.

So the original sentence is the most normal one for a beginner to use.

Normally, Esperanto includes subject pronouns like mi, vi, li, ŝi, ĝi, ni, ili.

So the usual form is:

• Ŝi skribas = she writes

Leaving out ŝi is generally not standard in normal Esperanto, because the verb ending -as only shows tense, not person. In other words, skribas by itself does not mean specifically I write, you write, or she writes.

Here is the breakdown:

• Ŝi = pronoun, she
• skribas = verb, present tense of skribi (to write)
• la = definite article, the
• nomon = noun nomo (name) + -n direct object ending
• de = preposition, of
• la = definite article, the
• filmo = noun, film

This is a useful example of a basic Esperanto sentence pattern:

subject + verb + direct object + prepositional phrase

A simple English-friendly guide:

• Ŝi ≈ shee
• skribas ≈ SKREE-bahs
• la ≈ lah
• nomon ≈ NOH-mon
• de ≈ deh
• la ≈ lah
• filmo ≈ FEEL-mo

A few important points:

• ŝ sounds like English sh
• stress in Esperanto is usually on the second-to-last syllable
  • skRIbas
  • NOmon
  • FILmo

So the full sentence sounds roughly like:

SHEE SKREE-bahs lah NOH-mon deh lah FEEL-mo

Sometimes Esperanto can use an adjective like filma (film-related), but that would not mean exactly the same thing here.

• la nomo de la filmo = the name of the film
• la filma nomo = the film-related name / cinematic name

For possession or close noun relationships like this, de is the normal and clearest choice.

So in this sentence, la nomo de la filmo is the natural way to say the name of the film.