После дождя земля в саду мокрая.

Because после requires the genitive case.

• dictionary form: дождь = rain
• genitive singular: дождя

So после дождя literally means after rain / after the rain.

Here is the case of each noun:

• дождя — genitive singular, because it follows после
• земля — nominative singular, because it is the subject
• в саду — a location form after в, traditionally called prepositional, with a special locative ending here

So the basic structure is:

• После дождя = time phrase
• земля = subject
• в саду = place
• мокрая = description of the subject

In Russian, the verb to be is usually not said in the present tense.

So:

• земля в саду мокрая = the ground in the garden is wet

But in the past or future, Russian does use a form of to be:

• После дождя земля в саду была мокрая. = The ground in the garden was wet after the rain.
• После дождя земля в саду будет мокрая. = The ground in the garden will be wet after the rain.

Because the adjective has to agree with земля.

Земля is:

• feminine
• singular
• nominative

So the adjective must also be:

• feminine singular nominative → мокрая

Compare:

• мокрый сад = a wet garden
• мокрая земля = wet ground
• мокрое окно = a wet window

Because some masculine nouns have a special form in -у / -ю after в or на when they mean location.

So:

• в саду = in the garden
• в лесу = in the forest
• на берегу = on the shore

But when you mean about something, you usually use the regular prepositional form:

• о саде = about the garden

So this contrast is very useful:

• в саду = location
• о саде = about the garden

Yes, мокра exists, but мокрая is more natural for most modern everyday Russian here.

• земля мокрая sounds neutral and common
• земля мокра sounds more literary, poetic, or stylistically marked

So for a learner, мокрая is the safer and more standard choice in normal speech.

Yes. Russian word order is more flexible than English word order.

This sentence starts with После дождя to set the scene first: after the rain.

Other orders are possible, for example:

• Земля в саду мокрая после дождя.
• В саду земля мокрая после дождя.

They all mean roughly the same thing, but the emphasis changes a little.

The original order, После дождя земля в саду мокрая, is very natural.

Russian has no articles.

So:

• земля can mean the ground, ground, or sometimes a piece of ground
• в саду can mean in the garden or in a garden

Usually the context tells you which meaning is intended.

That is why one Russian sentence can correspond to several English versions.

Земля can mean several things, including:

• earth
• ground
• soil
• land
• the planet Earth

In this sentence, because of в саду and мокрая, it clearly means ground/soil.

So here земля в саду мокрая means the ground in the garden is wet, not the Earth is wet.

The stressed syllables are:

• по́сле
• дождя́
• земля́
• саду́
• мо́края

So the sentence is pronounced approximately like this:

ПОС-ле дожд-Я зем-ЛЯ в са-ДУ МО-кра-я

A useful extra note: in fast speech, в before с can sound closer to f, so в саду may sound a bit like ф саду. That is normal Russian pronunciation.