ame no hi ha kansouki wo tukatte, aratta taoru wo hayaku tatamimasu.

の is linking two nouns here:

• 雨 = rain
• 日 = day

So 雨の日 literally means rain’s day, which in natural English becomes a rainy day or a day when it rains.

This is a very common pattern in Japanese:

• 夏の日 = a summer day
• 休みの日 = a day off
• 試験の日 = exam day

So 雨の日 is just a noun phrase meaning rainy day.

は makes 雨の日 the topic of the sentence:

• 雨の日は… = As for rainy days... / On rainy days...

This gives the sentence a general, habitual, or contrastive feeling. It suggests something like:

• On rainy days, I do this.
• Maybe on other kinds of days, I do something else.

If you said 雨の日に, that would simply mark the time on a rainy day, without the same topic/contrast nuance.

So:

• 雨の日は = On rainy days, ... / As for rainy days, ...
• 雨の日に = on a rainy day

In this sentence, は works well because it sounds like a routine or general habit.

使って is the て-form of 使う (to use).

Here, it connects one action to the next:

• 乾燥機を使って、洗ったタオルを早くたたみます。
• I use the dryer, and then quickly fold the washed towels.

The て-form often links actions like and, then, or by doing.

So 使って can suggest:

• use the dryer and...
• by using the dryer...

In this sentence, it mainly links the two actions in sequence.

Because there are two different verbs, and each verb has its own object.

1. 乾燥機を使って

   • 乾燥機 is the object of 使う
   • use the dryer

2. 洗ったタオルを早くたたみます

   • 洗ったタオル is the object of たたむ
   • fold the washed towels quickly

So the sentence has:

• dryer as the thing being used
• washed towels as the thing being folded

It is completely normal in Japanese to have more than one を in a sentence if there is more than one verb.

Because Japanese puts modifying information before the noun.

So:

• 洗ったタオル = the towels that were washed / washed towels

This is a relative clause in Japanese. Unlike English, Japanese does not need words like that, which, or who.

Compare:

• 洗ったタオル = the towel(s) I washed / the washed towel(s)
• 買った本 = the book I bought
• 作った料理 = the food I made

So 洗った directly describes タオル.

Because 洗った shows that the washing is already completed.

• 洗ったタオル = the towel(s) that have been washed
• 洗うタオル would mean something more like the towel(s) to wash or the towel(s) one washes, which is a different idea

So in this sentence, 洗ったタオル is natural because the towels have already been washed, and now the next action is folding them.

早く is the adverb form of 早い.

Here it modifies たたみます:

• 早くたたみます = fold quickly or fold early/soon

In many contexts, 早く can mean either:

• quickly
• early
• soon

The exact nuance depends on context.

In this sentence, it most naturally means that the folding is done quickly.
Grammatically, it is describing how the action of たたみます is done.

The subject is omitted because Japanese often leaves out information that is obvious from context.

So the sentence does not explicitly say:

• I
• we
• my family
• etc.

But in natural use, the listener usually understands it from the situation.

So this sentence could mean:

• I use the dryer and quickly fold the washed towels on rainy days
• or we use the dryer...

Japanese does this very often. A missing subject does not make the sentence incomplete.

たたみます is the polite form of たたむ (to fold).

So:

• たたむ = plain/dictionary form
• たたみます = polite form

The sentence is written in a polite style, which is very common in textbook Japanese and everyday conversation when speaking politely.

You can see the same thing with the other verbs:

• 使う → 使って (て-form of 使う)
• たたむ → たたみます
• 洗う → 洗った

Different forms are being used for different grammar purposes, but they all come from dictionary-form verbs.

Yes, that is the most natural reading.

• 乾燥機を使って、
• 洗ったタオルを早くたたみます。

The て-form of 使って usually links actions in order, so the sentence suggests:

1. use the dryer
2. fold the washed towels

It does not always have to mean a strict step-by-step sequence in every case, but here that is the natural interpretation.

So the sentence has the feel of a routine:

• On rainy days, I use the dryer and quickly fold the washed towels.

Most likely, yes.

乾燥機 literally means drying machine. Depending on context, it can refer to different kinds of dryers, but in a sentence about towels and rainy days, it normally means a clothes dryer.

So learners will usually understand:

• 乾燥機を使って = use the dryer

The context of laundry makes that meaning clear.