Jo oftere hun bruger vaskemaskinen, jo mindre vasketøj har hun på gulvet.

Jo ... jo is a common Danish pattern meaning the more ..., the ... or the more often ..., the less ....

So:

Jo oftere hun bruger vaskemaskinen, jo mindre vasketøj har hun på gulvet.

means:

The more often she uses the washing machine, the less laundry she has on the floor.

The first jo introduces one changing factor, and the second jo introduces the result that changes along with it.

Oftere is the comparative form of ofte (often), just like more often in English.

• ofte = often
• oftere = more often

Danish usually prefers this built-in comparative form rather than mere ofte.

So:

• Jo oftere ... = The more often ...

This works like:

• hurtigt → hurtigere = fast → faster
• ofte → oftere = often → more often

Because vasketøj is normally treated as an uncountable mass noun, like laundry in English.

You do not usually count vasketøj as separate items in this context. So Danish uses:

• mindre vasketøj = less laundry

not:

• færre vasketøj

Use færre with countable plural nouns, like:

• færre skjorter = fewer shirts
• færre sokker = fewer socks

But:

• mindre tøj
• mindre vasketøj

because these are being treated as amounts, not countable items.

This is because Danish follows the verb-second rule in main clauses.

In the second half of the sentence, the phrase jo mindre vasketøj comes first. When something other than the subject comes first, the finite verb must come before the subject:

• jo mindre vasketøj har hun på gulvet

not:

• jo mindre vasketøj hun har på gulvet

The normal main-clause pattern is:

• Hun har mindre vasketøj på gulvet.

But when jo mindre vasketøj is placed first, the verb moves in front of the subject:

• Jo mindre vasketøj har hun på gulvet.

Because each half is its own clause and needs its own subject.

• Jo oftere hun bruger vaskemaskinen
• jo mindre vasketøj har hun på gulvet

Even though both clauses refer to the same person, Danish still repeats hun. English does the same:

• The more often she uses the washing machine, the less laundry she has on the floor.

You cannot leave hun out here.

Vaskemaskinen means the washing machine. Danish often uses the definite form when referring to a specific thing that is understood from context, especially something in the home.

So here it means something like:

• the washing machine / her washing machine

Using en vaskemaskine would sound more like a washing machine, meaning any washing machine, not the one already understood in the situation.

Literally, på gulvet means on the floor.

In this sentence, it suggests dirty clothes are lying around on the floor. So the sentence implies:

• if she uses the washing machine more often,
• she leaves less laundry lying on the floor.

It is both literal and very natural in everyday Danish.

Grammatically, vasketøj is usually treated as a singular mass noun, even though it refers to many pieces of clothing.

It works a lot like English laundry:

• Vasketøjet er vådt. = The laundry is wet.
• Jeg har meget vasketøj. = I have a lot of laundry.

So even though it refers to multiple clothes, it behaves like one uncountable quantity.

Yes. In formal or slightly more literary Danish, you may also see:

• Jo oftere hun bruger vaskemaskinen, desto mindre vasketøj har hun på gulvet.

This means the same thing.

In modern everyday Danish, jo ... jo ... is very common and natural.

Yes. This is a very useful pattern:

Jo + comparative + clause, jo + comparative + clause

It means the more/less ..., the more/less ...

Examples:

• Jo mere du læser, jo bedre bliver du.
  = The more you read, the better you get.

• Jo længere vi venter, jo dyrere bliver det.
  = The longer we wait, the more expensive it becomes.

• Jo tidligere hun står op, jo mere når hun.
  = The earlier she gets up, the more she gets done.

So the sentence you were given is an example of a very productive grammar pattern.

No, not exactly. Danish jo has more than one use.

In this sentence, jo is part of the correlative structure jo ... jo ..., meaning the more ..., the ....

That is different from the separate discourse word jo, which can mean something like:

• after all
• as you know
• actually
• or an emphatic yes in reply to a negative question

So although it is the same word in form, its function here is grammatical, not conversational.