Après le sport, je mets mes serviettes dans le panier à linge, puis je démarre le sèche-linge.

Because après le sport is just a time expression, meaning after sport / after exercising. French often starts a sentence with this kind of phrase to set the scene.

• après = after
• le sport = sport, exercise, a workout

So Après le sport, ... means After working out, ... or After sports, ...

In context, a native English speaker should understand le sport here more as exercise / a workout session than as sports in general.

Because the verb is mettre and with je in the present tense, it becomes je mets.

Present-tense forms of mettre:

• je mets
• tu mets
• il/elle/on met
• nous mettons
• vous mettez
• ils/elles mettent

So je mets mes serviettes... means I put my towels...

A common learner issue is that je mets, tu mets, and il met sound very similar, but they are spelled differently.

French usually prefers a possessive adjective when talking about personal items, especially when it is natural to specify whose things they are.

So:

• mes serviettes = my towels

This sounds natural because the speaker is talking about their own towels after exercising.

Also:

• mes is used because serviettes is plural.
• Singular would be ma serviette.

It can mean either, depending on context.

• une serviette can be a towel
• une serviette can also be a napkin

Here, because the sentence talks about:

• after sport
• the laundry basket
• the dryer

the meaning is clearly towels.

Context is very important with this word.

Because dans means in / into, and that is the idea here: the towels are being put into the laundry basket.

• dans = in, inside, into
• le panier à linge = the laundry basket

So:

• je mets mes serviettes dans le panier à linge = I put my towels in the laundry basket

Using au here would not sound natural for this meaning.

Literally, panier à linge is something like basket for laundry / linen basket.

In French, à is often used in noun combinations to show purpose or function.

Examples:

• une tasse à café = a coffee cup
• une brosse à dents = a toothbrush
• un panier à linge = a laundry basket

So the à here has the idea of for.

Yes, et or ensuite could also work, but they are slightly different in feel.

• puis = then / next
• ensuite = then / afterwards
• et = and

Puis is good here because it clearly shows sequence: first one action happens, then the next one.

So:

• je mets mes serviettes dans le panier à linge, puis je démarre le sèche-linge

sounds like a natural chain of actions.

Compared with the others:

• et is simpler and less precise about sequence
• ensuite is also common, but puis is a bit neater and more compact here

Because démarrer means to start something, including a machine or appliance.

So:

• je démarre le sèche-linge = I start the dryer

This is very natural in French.

You might also hear:

• j’allume le sèche-linge = I turn on the dryer

But démarrer emphasizes starting the machine’s operation, which fits very well here.

Because sèche-linge is a fixed compound noun in French.

It is built from:

• sèche from sécher = to dry
• linge = laundry / linen

Together, un sèche-linge means a dryer or more literally a laundry-dryer.

Many common French compound nouns use hyphens, so this is something you often just have to learn as part of the word.

French often uses the definite article when the thing is already understood from the situation.

So le sèche-linge means the dryer, and it sounds natural because the speaker is talking about the dryer in their home or the relevant dryer in context.

French does not always use possessives as often as English does. Even if it is clearly my dryer, French may still simply say le sèche-linge.

It is in the present tense.

French often uses the present tense to describe:

• habitual actions
• routines
• step-by-step instructions
• what someone generally does

So this sentence can mean something like:

• After working out, I put my towels in the laundry basket, then I start the dryer

It may describe a routine, not only something happening right now.

A simple pronunciation guide is:

Après le sport, je mets mes serviettes dans le panier à linge, puis je démarre le sèche-linge.

Approximate English-style pronunciation: ah-pre luh spor, zhuh may may sair-vyet dahn luh pa-nyay ah lannzh, pwee zhuh day-mar luh sesh-lanzh

A few useful notes:

• Après: the è sounds like eh
• je sounds like zhuh
• mets sounds like may
• dans has a nasal vowel; the n is not fully pronounced like English n
• linge sounds roughly like lanzh
• puis sounds like pwee
• sèche-linge has sèche sounding like sesh

If you want to sound more natural, pay special attention to the nasal sound in dans and linge.