Det er tre år siden, jeg selv snublede på cykelstien og gik hjem med et forstuvet håndled.

This is a very common Danish structure for expressing how much time has passed since something happened.

• Det er tre år siden ... = It has been three years since ...
• Literally, it is close to It is three years since ...

So the sentence is framing the event by saying that it happened three years ago.

Yes, at can be left out here.

You can say both:

• Det er tre år siden, jeg selv snublede ...
• Det er tre år siden, at jeg selv snublede ...

Both are natural. Leaving out at is very common in everyday Danish.

The comma marks the start of the clause jeg selv snublede på cykelstien og gik hjem ...

In Danish, commas often show clause boundaries. In sentences like this, many writers put a comma before the following clause, especially in more traditional comma usage. So the comma is there to separate Det er tre år siden from the clause that explains since when.

Selv adds emphasis. Here it means something like myself.

So:

• jeg snublede = I tripped
• jeg selv snublede = I myself tripped

It can suggest emphasis such as:

• it really was me
• I personally experienced it
• I did it myself

It is not required for the basic meaning, but it adds focus.

Because the sentence describes a completed event in the past.

• snublede = past tense of snuble = tripped
• gik = past tense of gå = went / walked

Danish often uses the simple past for storytelling and for specific past events. Since this happened three years ago and is presented as a finished incident, the simple past is the natural choice.

På is the normal preposition here because the person was moving along or on the surface of the bike path.

• cykelsti = bike path / cycle lane
• på cykelstien = on the bike path

This does not mean over the bike path. It means the person was on that path when the fall happened.

Because Danish uses the definite form here: cykelstien = the bike path.

The ending -en is the definite article attached to the noun:

• en cykelsti = a bike path
• cykelstien = the bike path

In context, it refers to the specific bike path the speaker was on at that moment, so the definite form is natural.

Because hjem often works like an adverb meaning home or to home.

So:

• gå hjem = go home / walk home
• køre hjem = drive home
• komme hjem = come home

This is very common in Danish. You usually do not need a preposition or an article with hjem in expressions of motion.

Danish very often uses an indefinite noun phrase with body parts when the owner is already obvious from the context.

So:

• med et forstuvet håndled = with a sprained wrist

Even though English often prefers my wrist, Danish does not always need the possessive. The sentence already makes it clear that it is the speaker’s wrist.

Using mit would be possible, but it would sound more explicitly possessive or more emphatic than necessary here.

Because håndled is a neuter singular noun: et håndled.

In Danish, adjectives agree with the noun:

• en forstuvet ankel = a sprained ankle
• et forstuvet håndled = a sprained wrist
• de forstuvede håndled = the sprained wrists

So the -t in forstuvet appears because håndled is neuter singular and indefinite.

It describes the person’s condition.

In this sentence, med et forstuvet håndled means something like:

• while having a sprained wrist
• with a sprained wrist as the result

So gik hjem med et forstuvet håndled means the person walked home in that injured state. It does not mean they were carrying a wrist or taking one with them.