После дождя мои кеды стали мокрыми и грязными.

Because после requires the genitive case.

• дождь = nominative singular, the basic dictionary form
• дождя = genitive singular

So:

• после дождя = after the rain / after rain

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после фильма = after the movie

It can mean either, depending on context.

Russian has no articles like a or the, so дождя does not itself tell you whether it is rain or the rain. In this sentence, English will usually say after the rain or after it rained, because that sounds most natural.

So после дождя is literally just after rain, but in good English it often becomes:

• after the rain
• after it rained

In Russian, possessive words like мой must agree with the noun they describe in gender, number, and case.

Here, кеды is plural, so мой becomes мои.

Compare:

• мой ботинок = my shoe
• моя куртка = my jacket
• моё пальто = my coat
• мои кеды = my sneakers

English my never changes, but Russian possessives do.

Кеды means casual rubber-soled canvas shoes, often translated as:

• sneakers
• trainers
• tennis shoes
• sometimes plimsolls

The exact English word depends on region and context. In many learner translations, sneakers is the safest choice.

Usually, yes in normal everyday usage.

Кеды is commonly treated as a plural-only noun when talking about a pair of shoes, much like English often uses shoes rather than a shoe when speaking generally.

So you will often see:

• эти кеды
• новые кеды
• мои кеды

A singular form кед may occasionally appear in dictionaries or special contexts, but for learners, it is best to treat кеды as the normal form.

Because стать means to become, while быть means to be.

So:

• стали мокрыми и грязными = became wet and dirty
• были мокрыми и грязными = were wet and dirty

The sentence is describing a change of state: after the rain, the sneakers became wet and dirty. That is why стали is the natural choice.

Because after стать (to become), Russian usually puts the predicate adjective in the instrumental case.

So:

• стали мокрыми
• стали грязными

not normally:

• стали мокрые
• стали грязные

This is a very important pattern:

• Она стала счастливой. = She became happy.
• Он стал врачом. = He became a doctor.
• Они стали старыми. = They became old.

With стать, the thing someone/something becomes is typically in the instrumental.

Because they are:

• plural
• instrumental
• modifying the same noun: кеды

The nominative plural forms would be:

• мокрые
• грязные

But after стали, they change to instrumental plural:

• мокрыми
• грязными

So the ending reflects grammar, not meaning.

Стали is the past tense plural form of стать.

Here is the past tense of стать:

• стал = he became
• стала = she became
• стало = it became
• стали = they became

Because кеды is plural, the verb must also be plural:

• мои кеды стали...

In the Russian past tense, singular forms show gender, but plural forms do not.

Compare:

• ботинок стал мокрым = the shoe became wet
• куртка стала мокрой = the jacket became wet
• пальто стало мокрым = the coat became wet
• кеды стали мокрыми = the sneakers became wet

Once the subject is plural, Russian uses стали, with no gender distinction.

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

The neutral version is:

• После дождя мои кеды стали мокрыми и грязными.

But you could also hear:

• Мои кеды после дождя стали мокрыми и грязными.
• Мокрыми и грязными стали мои кеды после дождя.

These versions are grammatical, but they shift emphasis slightly. The original sentence sounds natural and neutral.

Russian often uses стать where English uses get.

So English:

• got wet
• got dirty

often becomes Russian:

• стал мокрым
• стал грязным

In the plural:

• стали мокрыми
• стали грязными

So стать is a very common way to express become / get with adjectives.

Yes, that would also be natural, but it means something slightly different in style.

Compare:

• стали мокрыми и грязными = became wet and dirty
• промокли и испачкались = got soaked and got dirty

The sentence you have is straightforward and descriptive. The alternative is more verbal and can sound a bit more dynamic. Both are natural, but they are not exact grammatical equivalents.