Nach dem Regen ist der Weg im Garten matschig.

Because nach is a preposition that takes the dative case in this meaning (after).

• der Regen = nominative
• dem Regen = dative

So:

• nach dem Regen = after the rain

This is something you usually just learn with the preposition:

• nach + dative

Examples:

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

German main clauses usually follow the V2 rule: the finite verb comes in the second position.

In this sentence, Nach dem Regen comes first, so the verb must come next:

• Nach dem Regen | ist | der Weg im Garten matschig

Even though der Weg is the subject, it comes after the verb because the first position is already taken by Nach dem Regen.

If you started with the subject instead, you could also say:

• Der Weg im Garten ist nach dem Regen matschig.

That is also correct, but the original version emphasizes when/under what conditions the path is muddy.

Der Weg is the subject of the sentence, so it is in the nominative case.

The verb ist goes with the subject:

• der Weg = the path
• ist = is

So the basic core is:

• der Weg ist matschig = the path is muddy

Everything else adds extra information:

• nach dem Regen = after the rain
• im Garten = in the garden

Im is a contraction of:

• in dem → im

So:

• im Garten = in the garden

This uses the dative case because it describes location rather than movement.

Compare:

• im Garten = in the garden (location, dative)
• in den Garten = into the garden (movement toward, accusative)

Here, the path is located in the garden, so German uses dative:

• in dem Garten / im Garten

Because matschig is being used as a predicate adjective, not directly before a noun.

In German:

• Attributive adjective (before a noun) takes endings:
  • ein matschiger Weg = a muddy path
• Predicate adjective (after sein, werden, bleiben) does not take an ending:
  • Der Weg ist matschig. = The path is muddy.

So here:

• ist matschig = is muddy

No adjective ending is used after ist.

Matschig means something like:

• muddy
• slushy
• soft and wet from mud

In this sentence, it means the path has become wet and muddy after the rain.

It comes from Matsch, meaning mud or slush.

A learner should notice that matschig is a very natural adjective for ground, paths, shoes, fields, etc. after rain.

In normal understanding, im Garten most naturally describes Weg:

• der Weg im Garten = the path in the garden

So the sentence is understood as:

• The path in the garden is muddy after the rain.

Grammatically, adverbial phrases can sometimes feel flexible, but here most people will hear im Garten as specifying which path.

Yes, that is grammatically possible, but it sounds a bit less natural in many contexts.

The original sentence:

• Nach dem Regen ist der Weg im Garten matschig.

is very natural because it starts with the time/circumstance and then gives the main statement.

A clearer alternative is:

• Der Weg im Garten ist nach dem Regen matschig.

That keeps der Weg im Garten together as one noun phrase.

German word order is flexible, but some orders sound smoother than others.

Because matschig is an adjective, not a noun.

The structure is:

• der Weg = the path
• ist = is
• matschig = muddy

German is simply saying:

• The path is muddy.

If you wanted a noun instead, you would need a different structure, but that is not what is happening here.

Yes. Nach has several common meanings, including:

• after
  
  • nach dem Regen = after the rain
• to (with many countries/cities)
  • nach Berlin = to Berlin
• according to
  • nach dem Bericht = according to the report

In this sentence, the meaning is clearly after because it refers to time.

If you wanted to say the paths in the garden are muddy after the rain, you would say:

• Nach dem Regen sind die Wege im Garten matschig.

Changes:

• der Weg → die Wege (plural)
• ist → sind

Notice that matschig still does not change, because it is still a predicate adjective after sein.

Yes, Regen is masculine:

• der Regen

Its forms are:

• nominative: der Regen
• accusative: den Regen
• dative: dem Regen
• genitive: des Regens

That is why nach dem Regen uses dem.

Usually, no—not in standard German.

You normally say:

• nach dem Regen

because Regen here refers to a specific rain event or rainfall as a situation, and German normally uses the article in this phrase.

So nach dem Regen is the idiomatic choice.

A helpful way to break it down is:

• Nach dem Regen = time/circumstance
• ist = verb
• der Weg im Garten = subject
• matschig = predicate adjective

So the core pattern is:

• [Time phrase] + [verb] + [subject] + [adjective]

This is a very common German sentence pattern, especially when something other than the subject comes first.