Læreren siger, at denne regel også gælder i morgen.

Because Danish usually puts the definite article at the end of the noun.

• en lærer = a teacher
• læreren = the teacher

So læreren siger means the teacher says.

This is different from English, where the comes before the noun.

Here at means that.

So:

• Læreren siger, at ... = The teacher says that ...

It introduces a subordinate clause, just like English that does.

In some spoken Danish, at may sometimes be omitted, but in a sentence like this it is completely normal and standard to include it.

Because denne means this, and when you use this/that in Danish, the noun normally appears in its indefinite form.

So:

• reglen = the rule
• denne regel = this rule

This is similar to English: we say this rule, not this the rule.

Because regel is a common gender noun in Danish.

• en regel = a rule
• so with this, you use denne

Basic pattern:

• denne
  • common gender singular noun
• dette
  • neuter singular noun

For example:

• denne regel = this rule
• dette hus = this house

Gælder is the present tense of at gælde.

In this sentence, it means something like:

• applies
• is valid
• is in effect
• counts

So denne regel også gælder i morgen means that the rule also applies tomorrow.

Because this is a subordinate clause introduced by at.

In Danish subordinate clauses, words like også, ikke, altid, etc. often come before the finite verb.

So:

• at denne regel også gælder i morgen
  = subordinate clause word order

Compare that with a main clause:

• Denne regel gælder også i morgen.
  = main clause word order

This is a very important Danish pattern:

• main clause: verb usually comes early
• subordinate clause: adverbs often come before the verb

It would sound less natural in standard Danish.

The normal order in a subordinate clause is:

• subject + adverb + verb

So the usual version is:

• at denne regel også gælder i morgen

Putting også after the verb is much more natural in a main clause:

• Denne regel gælder også i morgen.

Yes. Siger is the present tense of at sige = to say.

So:

• jeg siger = I say / I am saying
• læreren siger = the teacher says / is saying

Danish present tense often covers both simple present and present progressive meanings, depending on context.

Because Danish, like English, can use the present tense for something connected to the future when the time is clear from context.

Here:

• gælder i morgen literally looks like applies tomorrow

That is completely normal. The time expression i morgen already tells you that it is about tomorrow, so Danish does not need a special future form here.

I morgen means tomorrow.

In modern Danish it is normally written as two words:

• i morgen

You should usually treat it as a fixed time expression meaning tomorrow.

It breaks down like this:

• Læreren = the teacher
• siger = says
• at = that
• denne regel = this rule
• også = also
• gælder = applies / is valid
• i morgen = tomorrow

So the structure is:

• main clause: Læreren siger
• subordinate clause: at denne regel også gælder i morgen

This is a very common Danish sentence pattern:
[someone] says/thinks/knows + at + clause

A rough English-friendly approximation is:

LAIR-er-en SEE-er, at DEN-uh RAI-el AW-so GEL-er ee MOR-en

A few notes:

• æ in lærer and gælder is a front vowel that does not match English perfectly.
• gælder is pronounced more like GEL-er than how it looks.
• morgen is often pronounced something like MOR-en.
• denne sounds roughly like DEN-uh.

This is only approximate, but it can help you recognize the sentence when you hear it.