Jeg tager handsker på, når jeg cykler til arbejdet om vinteren.

Because tage ... på is a particle verb in Danish, and here it means to put on clothing.

So:

• tage på
  • clothing item = put on
• Jeg tager handsker på = I put on gloves

With a noun object like handsker, the object usually comes between the verb and the particle:

• Jeg tager handsker på
• Hun tager jakken på

With a pronoun, Danish also keeps the pronoun before på:

• Jeg tager dem på = I put them on

So even though English keeps put on together more clearly, Danish often splits this kind of verb.

Because når means when/whenever in the sense of something that regularly happens, while hvis means if.

In this sentence, the idea is habitual:

• Når jeg cykler til arbejdet om vinteren = When/whenever I bike to work in winter

That means this is something that happens repeatedly.

Compare:

• Når jeg cykler til arbejdet om vinteren, tager jeg handsker på
  = Every time this happens, I put on gloves.
• Hvis jeg cykler til arbejdet om vinteren, tager jeg handsker på
  = If I happen to bike to work in winter, then I put on gloves.

So når is the natural choice for a repeated routine.

Because handsker is an indefinite plural noun here, just like English gloves in I put on gloves.

You are not talking about specific gloves already identified as the gloves. You just mean gloves in general.

Compare:

• Jeg tager handsker på = I put on gloves
• Jeg tager mine handsker på = I put on my gloves
• Jeg tager de handsker på, som du gav mig = I put on the gloves that you gave me

So the bare plural handsker is completely normal.

Because Danish often uses the definite form where English uses no article.

So:

• arbejdet literally looks like the work
• but in context til arbejdet often means to work or to the workplace

This is one of those places where Danish and English do not match word-for-word.

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

• et arbejde = a job / a piece of work
• arbejdet = the job / the work

Here it refers to the speaker’s known destination: their workplace.

Yes. In fact, på arbejde is very common in Danish when talking about going to work or being at work.

For example:

• Jeg cykler på arbejde = I bike to work
• Jeg er på arbejde = I am at work

The sentence you were given, cykler til arbejdet, is understandable and grammatical, and it focuses more directly on the destination. But many Danes would very naturally say:

• Jeg tager handsker på, når jeg cykler på arbejde om vinteren.

So both can work, but på arbejde is often the more idiomatic commuting expression.

Because Danish uses the present tense for habitual actions, just as English often does.

This sentence is not describing one single event happening right now. It describes a routine:

• Jeg tager handsker på = I put on gloves / I wear gloves
• når jeg cykler ... = when I bike ...

So the present tense here means something like whenever this situation happens.

This is very normal in Danish:

• Jeg drikker kaffe om morgenen = I drink coffee in the morning
• Hun løber, når vejret er godt = She runs when the weather is good

Om vinteren means in winter or during the winter season in general.

In Danish, om is often used with parts of the day, days, and seasons when talking about something that happens generally or repeatedly:

• om morgenen = in the morning
• om sommeren = in the summer
• om vinteren = in the winter

A useful contrast is:

• om vinteren = in winter / during winters generally
• i vinter = this winter

So in your sentence, om vinteren fits because the meaning is habitual and general, not just one specific winter.

Because når jeg cykler til arbejdet om vinteren is a subordinate clause, and subordinate clauses usually keep subject + verb order in Danish.

So:

• når jeg cykler = correct
• når cykler jeg = incorrect

This is different from a main clause, where Danish often follows verb-second word order.

Compare:

• Jeg cykler til arbejdet om vinteren = main clause
• når jeg cykler til arbejdet om vinteren = subordinate clause

So after når, you normally get regular subject-verb order.

Then the main clause changes word order because Danish main clauses follow the verb-second rule.

So you can say:

• Når jeg cykler til arbejdet om vinteren, tager jeg handsker på.

Notice what happens in the main clause:

• tager jeg
• not jeg tager

Why? Because the whole fronted clause Når jeg cykler til arbejdet om vinteren takes the first position, so the finite verb tager must come next.

That is a very important Danish pattern:

• I dag tager jeg handsker på
• Om vinteren tager jeg handsker på
• Når jeg cykler til arbejdet om vinteren, tager jeg handsker på

Because når jeg cykler til arbejdet om vinteren is a subordinate clause, and many Danish writers put a comma before subordinate clauses.

So:

• Jeg tager handsker på, når jeg cykler til arbejdet om vinteren.

That comma helps show the clause boundary.

A small extra detail: modern Danish allows two comma styles, so you may also see the sentence written without that comma:

• Jeg tager handsker på når jeg cykler til arbejdet om vinteren.

Both can be correct depending on the comma system being used.

Because tage ... på and have ... på mean slightly different things.

• tage handsker på = put on gloves
  This focuses on the action of putting them on.
• have handsker på = have gloves on / be wearing gloves
  This focuses on the state of wearing them.

So your sentence emphasizes the action the speaker does when that situation occurs.

Compare:

• Jeg tager handsker på, når jeg cykler til arbejdet om vinteren.
  = I put on gloves when I bike to work in winter.
• Jeg har handsker på, når jeg cykler til arbejdet om vinteren.
  = I wear gloves when I bike to work in winter.

Both are possible, but they highlight different things.