Learn Hardcore Hebrew: אחרי הגשם יש הרבה בוץ בגינה, ולכן הילדים לא רצו לשבת שם. - After the rain there is a lot of mud in the garden, so the children did not want to sit there.

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.

Why is it הגשם and not just גשם?

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.

What does יש mean here?

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.