Jeg tager mit forklæde på, så jeg ikke får mel på tøjet.

Because tage ... på is a very common Danish verb expression meaning to put on clothing.

In this sentence, tager is the finite verb, and på is a particle that stays later in the clause:

Jeg tager mit forklæde på.

This is normal in Danish. English does something similar with put on:

• I put my apron on.

So på is not random here; it belongs with tage.

Because forklæde is a neuter noun.

The basic form is:

• et forklæde = an apron

Danish possessives change according to the noun:

• min for common gender singular
• mit for neuter singular
• mine for plural

So:

• min jakke = my jacket
• mit forklæde = my apron
• mine sko = my shoes

Because Danish does not use an article together with a possessive like min/mit/mine.

So you say:

• mit forklæde = my apron

not:

• mit et forklæde

This works just like English, where you say my apron, not my an apron.

Here så means so that or so in the sense of with the result/purpose that.

So:

• Jeg tager mit forklæde på, så jeg ikke får mel på tøjet.

means:

• I put on my apron so that I don’t get flour on my clothes.

This is not the same as så meaning then.

Because så introduces a subordinate clause here, and subordinate clauses in Danish have different word order.

In a subordinate clause, ikke usually comes before the finite verb:

• så jeg ikke får mel på tøjet

Compare that with a main clause:

• Jeg får ikke mel på tøjet.

So the contrast is:

• main clause: jeg får ikke
• subordinate clause: jeg ikke får

This is one of the most important word-order patterns in Danish.

Because få is the normal verb for get in expressions like get something on something.

So:

• få mel på tøjet = get flour on your clothes
• få kaffe på skjorten = get coffee on your shirt

It is very idiomatic Danish. English uses get the same way, so this part matches English quite well.

Because mel is a mass noun, like flour in English.

When you mean flour in a general sense, you usually do not use an article:

• mel = flour

That is similar to English:

• I don’t want flour on my clothes

not:

• a flour

If you were talking about specific flour, Danish could use something else, such as:

• melet = the flour
• noget mel = some flour
• det mel = that flour

Both are possible, but på tøjet sounds very natural here.

Tøj means clothes/clothing, and tøjet is the definite form: the clothes. In context, that naturally means the clothes I’m wearing or simply my clothes.

So:

• på tøjet = on the clothes / on my clothes

Using mit tøj would make the possession more explicit, but it is not necessary because it is already obvious whose clothes we mean.

Because tøj is a collective noun in Danish.

Grammatically, it behaves like a singular noun, even though in English we usually translate it as clothes or clothing.

So you get forms like:

• et tøj
• tøjet

This is similar to how English clothing is grammatically singular even though it refers to many items.

This is the difference between an action and a state.

• Jeg tager mit forklæde på = I put on / am putting on my apron
• Jeg har mit forklæde på = I have my apron on / I am wearing my apron

So in your sentence, tager ... på is used because the speaker is talking about the action of putting it on.

Because the sentence has two clauses:

• Jeg tager mit forklæde på
• så jeg ikke får mel på tøjet

In standard written Danish, clauses are normally separated with a comma, so the comma before så is expected.

It helps show that the second part is a separate clause explaining the purpose or result.