Nach dem Regen ist die Windschutzscheibe schmutzig, darum macht sie die Scheibenwischer an.

Nach dem Regen means after the rain.

The preposition nach usually takes the dative case when it means after in a time sense.
That is why der Regen becomes dem Regen.

• der Regen = the rain
• nach dem Regen = after the rain

So this is a very common pattern:

• nach dem Essen = after the meal
• nach der Arbeit = after work

Windschutzscheibe is a feminine noun, so its article in the nominative singular is die.

• die Windschutzscheibe = the windshield

In this sentence, it is the subject of the first clause:

• Die Windschutzscheibe ist schmutzig.

German nouns have grammatical gender, so you simply have to learn Windschutzscheibe as feminine.

It is a compound noun, which is very common in German.

It is made up of:

• Wind = wind
• Schutz = protection
• Scheibe = pane / glass panel

So literally it is something like wind-protection glass, which corresponds to windshield in English.

Because schmutzig is a predicate adjective, not an adjective directly before a noun.

In German:

• Before a noun, adjectives usually take endings:
  
  • die schmutzige Windschutzscheibe
• After sein, werden, bleiben, etc., adjectives usually do not take endings:
  
  • Die Windschutzscheibe ist schmutzig.

So here:

• ist schmutzig = is dirty

Darum means therefore, that’s why, or for that reason.

It connects the first idea to the second:

• The windshield is dirty
• therefore she turns on the windshield wipers

So darum signals cause and result.

Similar words are:

• deshalb
• deswegen

All of these are very common in German.

Because German main clauses follow the verb-second rule.

When darum is placed first, it counts as the first element of the clause, so the conjugated verb must come next:

• Darum macht sie die Scheibenwischer an.

Structure:

• darum = position 1
• macht = position 2
• sie = after the verb

This is called inversion from an English learner’s point of view.

Compare:

• Sie macht die Scheibenwischer an.
• Darum macht sie die Scheibenwischer an.

Because the verb is anmachen, a separable verb.

In a main clause, separable verbs split:

• anmachen → macht ... an

So:

• Sie macht die Scheibenwischer an. = She turns on the windshield wipers.

Other examples:

• Ich stehe um 7 Uhr auf. = I get up at 7.
• Er ruft seine Freundin an. = He calls his girlfriend.

In the infinitive, it stays together:

• die Scheibenwischer anmachen

Here anmachen means to switch on or to turn on.

Yes, another very common verb would be einschalten:

• Sie schaltet die Scheibenwischer ein.

That is often a bit more neutral or technical.
Anmachen is common in everyday speech.

So both can work:

• die Scheibenwischer anmachen
• die Scheibenwischer einschalten

Here die is the plural definite article: the.

• der Scheibenwischer = the windshield wiper
• die Scheibenwischer = the windshield wipers

So this is not feminine singular here; it is plural.

German uses die for:

• feminine singular nominative/accusative
• plural nominative/accusative

You have to tell from the noun and the context which one it is.

This is another compound noun:

• Scheibe = pane / window / glass
• Wischer = wiper

So Scheibenwischer literally means window-glass wiper, i.e. windshield wiper.

In everyday German, this is the normal word for the car’s wipers.

This is a very good question, because sie can mean several things in German depending on context:

• she
• it for a feminine noun
• they

Here, the intended meaning is she.

Grammatically, sie could in theory refer back to die Windschutzscheibe, because that noun is feminine. But the context makes that unlikely, because a windshield does not normally turn on windshield wipers by itself.

So in this sentence, a learner should understand sie as a female person.

This is one reason German pronouns can sometimes feel more ambiguous than English ones when seen on the page by themselves.

Yes, in normal punctuation it is appropriate here because you have two main clauses:

1. Nach dem Regen ist die Windschutzscheibe schmutzig
2. darum macht sie die Scheibenwischer an

They are being linked in one sentence, so the comma separates the two clauses.

You could also write them as two separate sentences:

• Nach dem Regen ist die Windschutzscheibe schmutzig. Darum macht sie die Scheibenwischer an.

Again, this is the German verb-second rule.

The first clause begins with Nach dem Regen, which takes the first position.
So the conjugated verb ist must come next:

• Nach dem Regen = position 1
• ist = position 2
• die Windschutzscheibe = after the verb

Compare the more basic order:

• Die Windschutzscheibe ist nach dem Regen schmutzig.

Both are possible, but starting with Nach dem Regen emphasizes the time phrase.

Yes. Dreckig and schmutzig can both mean dirty.

In many everyday situations:

• schmutzig = dirty
• dreckig = dirty / grubby / messy

Schmutzig often sounds a bit more neutral.
Dreckig can sound a little more colloquial or stronger depending on context.

So your version is understandable and natural.

Yes, grammatically it is fine.

That said, depending on context, a native speaker might also say something a little more idiomatic, such as:

• Nach dem Regen ist die Windschutzscheibe nass, darum macht sie den Scheibenwischer an.
• Die Windschutzscheibe ist schmutzig, also macht sie die Scheibenwischer an.

One small real-world note: after rain, the windshield may be wet or spotted, not necessarily dirty. But as a grammar example, the sentence works well.

Because the sentence is talking about the windshield wipers in the plural.

• den Scheibenwischer = the windshield wiper (singular, masculine accusative)
• die Scheibenwischer = the windshield wipers (plural accusative or nominative)

Cars normally have two wipers, so the plural is natural here.

If you were talking about just one wiper, you would say:

• Sie macht den Scheibenwischer an.

But that would be less usual in this context.