Der Film, von dem meine Schwester seit Tagen schwärmt, läuft heute Abend im Kino.

It has a main clause with a relative clause inserted into it.

• Main clause: Der Film läuft heute Abend im Kino.
• Relative clause: von dem meine Schwester seit Tagen schwärmt

The relative clause gives extra information about der Film.

So the full sentence is basically:

• The film ... is showing at the cinema tonight
• with the added detail
• ... that my sister has been raving about for days

Because relative clauses in German are normally set off with commas.

So whenever you add a clause that describes a noun, German usually marks it with commas:

• Der Mann, der dort steht, ...
• Das Buch, das ich lese, ...
• Der Film, von dem ... schwärmt, ...

This is much more regular in German than in English.

Von dem is a relative pronoun phrase referring back to der Film.

Literally, it looks like from/of which, but in natural English that sounds awkward. Here it works more like:

• that my sister has been raving about
• or which my sister has been raving about

The reason it uses von is that the verb schwärmen often goes with von + dative.

Because von always takes the dative case.

The noun being referred to is der Film, which is masculine singular, so the matching dative relative pronoun is dem.

A useful way to think about it:

• The noun is der Film
• Inside the relative clause, the verb/preposition pattern is von + dative
• Therefore: von dem

So even though Film is der in the main clause, it becomes dem here because the case is determined by its role inside the relative clause, not by the case of the original noun.

Because German verbs often require their own specific prepositions, and they do not always match English ones.

Here, the idiomatic pattern is:

• von etwas schwärmen = to rave about something

So German says, literally, something like to enthuse of/about something, using von.

This is very common in German: you cannot always translate the preposition word-for-word from English.

Here schwärmen means:

• to rave
• to gush
• to speak very enthusiastically

So meine Schwester schwärmt von dem Film means that the sister is talking about the film with a lot of enthusiasm.

Be aware that schwärmen can have other meanings in other contexts, but here it clearly means enthusiastic praise.

Because relative clauses are a type of subordinate clause, and in German subordinate clauses send the conjugated verb to the end.

Compare:

• Main clause: Meine Schwester schwärmt seit Tagen von dem Film.
• Relative clause: ..., von dem meine Schwester seit Tagen schwärmt, ...

That final-verb position is one of the most important word-order patterns in German.

This is a very common difference between German and English.

With seit, German normally uses the present tense for something that started in the past and is still continuing now.

So:

• Meine Schwester schwärmt seit Tagen ...

corresponds to English:

• My sister has been raving for days ...

So even though German uses the present tense form schwärmt, the meaning is often what English expresses with the present perfect continuous.

Seit Tagen means for days or for days now.

A few useful points:

• seit = since / for
• It takes the dative
• Tagen is the dative plural of Tage

So:

• seit Tagen = for days
• seit einer Woche = for a week
• seit gestern = since yesterday

Yes, laufen often means to run, but it also has other common meanings depending on context.

With films, TV shows, plays, and similar things, laufen often means:

• to be on
• to be showing
• to be playing

So:

• Der Film läuft heute Abend im Kino
  means
• The film is showing at the cinema tonight

This is a very natural use of laufen in German.

Because im Kino describes a location, not movement.

• im Kino = in dem Kino = in the cinema
• ins Kino = in das Kino = into the cinema / to the cinema

Here the film is being shown at that location, so German uses the dative form:

• im Kino

If a person were going there, you would often use ins Kino:

• Wir gehen heute Abend ins Kino.

Because it describes the film, so it is placed directly after the noun it modifies.

German usually puts a relative clause immediately after the noun it belongs to:

• Der Film, von dem ...
• Die Frau, die ...
• Das Buch, das ...

That makes it clear what the relative clause refers to. Here, von dem meine Schwester seit Tagen schwärmt clearly belongs to Der Film.

Yes. A very natural version would be:

• Meine Schwester schwärmt seit Tagen von dem Film. Der Film läuft heute Abend im Kino.

Or combined differently:

• Der Film läuft heute Abend im Kino. Meine Schwester schwärmt seit Tagen von ihm.

The version with the relative clause is more elegant and connected, but the meaning stays the same.