Jeg holder begge hænder på styret, når jeg cykler hjem.

Holde means to hold / keep / maintain in a position, while have simply means to have.

So:

• Jeg har begge hænder på styret = I have both hands on the handlebars
• Jeg holder begge hænder på styret = I keep/hold both hands on the handlebars

In this sentence, holder emphasizes that the hands are being kept there while riding.

In Danish, body parts often do not need a possessive like my, your, etc. when it is already obvious whose body parts they are.

So Danish often says:

• Jeg vasker hænder = I wash my hands
• Han løfter armen = He lifts his arm
• Jeg holder begge hænder på styret = I keep both hands on the handlebars

You can add a possessive for emphasis, for example begge mine hænder, but it is less neutral here. The plain begge hænder sounds natural.

Begge means both.

It is used when there are two of something:

• begge hænder = both hands
• begge børn = both children

It does not change form here. You just put it before the noun.

Because the sentence is talking about both hands, so the noun must be plural.

The singular is:

• en hånd = a hand

The plural is:

• hænder = hands

This is an irregular plural, so it is not formed by simply adding -er to hånd in a predictable way.

The base noun is et styr or, more commonly in this meaning, styret as the definite form referring to the handlebars/steering bar.

Here, styret means the handlebars or the handlebar area. Danish often uses the singular definite form for something English expresses as a plural:

• på styret = on the handlebars

So even though English says the handlebars, Danish uses styret.

Because på describes a location/position, not movement toward something.

• på styret = on the handlebars
• til styret would mean to the handlebars, which suggests movement toward them

Since the hands are already resting there, på is the natural choice.

In standard Danish, a comma is usually written before a subordinate clause, and når jeg cykler hjem is a subordinate clause.

So the sentence is divided like this:

• Jeg holder begge hænder på styret = main clause
• når jeg cykler hjem = subordinate clause

That is why there is a comma before når.

Here når means when in the sense of whenever or at the time that, especially for something habitual or repeated.

• Når jeg cykler hjem = When(ever) I cycle home

Use når for repeated or general situations.

Da is usually used for a specific event in the past:

• Da jeg cyklede hjem i går, regnede det = When I was cycling home yesterday, it rained

So in your sentence, når is correct because it sounds like a general habit.

Because after jeg in a normal present-tense statement, Danish uses the present form of the verb.

The infinitive is:

• at cykle = to cycle

The present tense is:

• jeg cykler = I cycle / I am cycling

So:

• når jeg cykler hjem = when I cycle home

Hjem means home and often works like an adverb in Danish, showing direction.

• Jeg går hjem = I go home
• Hun kører hjem = She drives home
• Jeg cykler hjem = I cycle home

Notice that Danish usually does not need a preposition here. English also says go home, not go to home.

Yes. You can say:

• Når jeg cykler hjem, holder jeg begge hænder på styret.

That means the same thing.

But notice the word order change in the main clause after the subordinate clause comes first:

• Når jeg cykler hjem, holder jeg ...
• not Når jeg cykler hjem, jeg holder ...

This happens because Danish is a V2 language: in a main clause, the finite verb normally comes in second position.

It specifically means go home by bike / cycle home.

• cykle = to cycle
• hjem = home

So jeg cykler hjem means that the speaker is travelling home on a bicycle.

Sometimes yes, depending on context.

Danish present tense often covers both:

• I hold
• I am holding

So Jeg holder begge hænder på styret can mean:

• I keep both hands on the handlebars
• I’m holding both hands on the handlebars

In natural English, though, I keep both hands on the handlebars when I cycle home is usually the best translation.

Most naturally, it describes a habit or general behavior:

• Jeg holder begge hænder på styret, når jeg cykler hjem.
• I keep both hands on the handlebars when I cycle home.

Because of når plus present tense, it sounds like something the speaker generally does whenever they ride home. It could be understood in a present situation too, but the habitual reading is the most natural.