Mit dem Resturlaub darf ich mir den Freitag freinehmen.

Because mit always takes the dative case.

• Base form: der Resturlaub
• After mit: mit dem Resturlaub

So this is not a special idiom of this sentence; it is just normal case grammar after mit.

Resturlaub means your remaining vacation entitlement or unused leave days.

It is made from:

• Rest = remainder / what is left
• Urlaub = vacation / leave

So Resturlaub is not a different kind of holiday; it means the vacation days you still have left to use.

Darf expresses permission, authorization, or what is allowed under the rules.

So here it suggests something like:

• I’m allowed to take Friday off
• I’m entitled to take Friday off
• My remaining leave makes it possible from a policy point of view

If you used kann, it would sound more like ability or practical possibility rather than permission.

Mir is the dative reflexive pronoun of ich.

The verb is often used as sich freinehmen, especially when talking about taking time off for oneself:

• ich nehme mir den Freitag frei
• du nimmst dir den Freitag frei
• er nimmt sich den Freitag frei

English usually does not translate this reflexive idea directly, but German often includes it with this verb. It makes the expression sound natural and idiomatic.

Because Freitag is the direct object of freinehmen, so it is in the accusative case.

Also, weekdays are masculine nouns in German:

• der Freitag = nominative
• den Freitag = accusative

So the structure is basically:

• take Friday off

German treats Friday as the thing being taken off.

Because den Freitag and am Freitag do different jobs.

• den Freitag = the day you are taking off
• am Freitag = on Friday, just giving the time

Compare:

• Ich nehme mir den Freitag frei.
  = I’m taking Friday off.

• Ich habe am Freitag frei.
  = I’m off on Friday.

So den Freitag fits the verb freinehmen much better here.

Freinehmen is a separable verb.

That means:

• In the infinitive, it appears as freinehmen
• In a normal main clause, it splits:
  • Ich nehme mir den Freitag frei.
• With a modal verb like dürfen, the infinitive stays together at the end:
  • Ich darf mir den Freitag freinehmen.

So the sentence structure is completely normal for a separable verb used with a modal.

German main clauses usually follow the verb-second rule.

That means the finite verb must come in the second position. Here:

1. Mit dem Resturlaub = first position
2. darf = second position

Then the rest follows:

• ich
• mir
• den Freitag
• freinehmen

So even though the sentence starts with a prepositional phrase, the verb still has to stay in second position.

Using den Freitag makes it sound like a specific Friday, usually the one that is relevant in the situation.

German often uses the article with weekdays when they are treated as nouns in a sentence like this.

• den Freitag freinehmen = take that Friday off

If you dropped the article, it would sound less standard here. The version with den is the more natural one in this structure.

They are similar, but not exactly the same.

• Urlaub nehmen = to take vacation/leave
• sich freinehmen = to take time off / take a day off for oneself

Urlaub nehmen is often more general or more formal.
Sich freinehmen is very common when talking about a specific day or short period:

• Ich nehme nächste Woche Urlaub.
• Ich nehme mir den Freitag frei.

So in this sentence, sich freinehmen sounds especially natural because it is about taking off one particular day.

You may see both in real life, but learners should especially recognize that it behaves like a separable verb:

• infinitive: freinehmen
• split form: Ich nehme mir den Freitag frei.

So even if you come across frei nehmen, the important grammatical point is that in a normal finite clause, frei moves to the end.