אחרי הגשם יש הרבה בוץ בגינה, ולכן הילדים לא רצו לשבת שם.

אחרי means after. So אחרי הגשם means after the rain.

This whole phrase sets the time for the sentence, like saying After the rain, ... in English. Hebrew often puts time phrases at the beginning of a sentence this way.

The ה at the beginning is the definite article, so הגשם means the rain.

Here it refers to a specific rainfall that is understood from the context, not just rain in general. So אחרי הגשם is more like after the rain or after it rained, not simply after rain in an abstract sense.

יש is the Hebrew word used to express existence, so it means there is or there are.

In this sentence:

יש הרבה בוץ בגינה = There is a lot of mud in the garden

A very important point for learners: Hebrew uses יש for both singular and plural.

• יש ילד = there is a child
• יש ילדים = there are children

So יש does not change the way English is/are does.

הרבה means a lot of, much, or many.

Since בוץ means mud, which is an uncountable noun, English would normally say a lot of mud or much mud, not many mud. Hebrew still uses הרבה very naturally with mass nouns:

• הרבה מים = a lot of water
• הרבה חול = a lot of sand
• הרבה בוץ = a lot of mud

So הרבה works with both countable and uncountable nouns.

In normal unpointed Hebrew spelling, בגינה can represent either in a garden or in the garden. Context tells you which one is meant.

Here, the most natural translation is in the garden, because the sentence seems to refer to a specific garden already known to the speaker and listener.

Also, Hebrew often combines the preposition ב (in) with the definite article ה (the), so ב + הגינה becomes בגינה.

ולכן means and therefore, and so, or simply so.

It is made of:

• ו = and
• לכן = therefore

So the sentence structure is:

After the rain there is a lot of mud in the garden, and therefore / so the children did not want to sit there.

It is a very useful connector, and it sounds a bit more formal or written than a simple conversational אז in some contexts.

הילדים means the children.

• ילדים = children
• הילדים = the children

The ה again is the definite article the.

Also, ילדים is grammatically masculine plural. In Hebrew, masculine plural can refer to:

• a group of boys
• a mixed group
• children in general when the group is not specified as all female

If the sentence were specifically about girls, you would expect הילדות.

רצו is the past-tense plural form of לרצות, which means to want.

Because the subject is הילדים (the children), the verb must also be plural:

• הילד רצה = the boy wanted
• הילדה רצתה = the girl wanted
• הילדים רצו = the children wanted

So לא רצו means did not want.

In Hebrew, לא usually goes before the finite verb, the verb that is actually carrying the tense.

Here the main verb is רצו = wanted, and לשבת is an infinitive = to sit.

So:

• רצו לשבת = wanted to sit
• לא רצו לשבת = did not want to sit

This is the normal Hebrew pattern with verbs like to want, to need, to be able, and so on:
main verb + infinitive

לשבת is the infinitive of ישב, and here it means to sit.

So רצו לשבת literally means wanted to sit.

Depending on context, לשבת can sometimes have broader meanings like to stay seated or even to live/dwell, but in this sentence the meaning is clearly just to sit.

שם can mean either there or name.

In this sentence, it means there, because it comes after לשבת:

לשבת שם = to sit there

The meaning name would not make sense in this context. Hebrew often relies on context to tell you which meaning is intended.