Après la réunion, je remets le badge dans mon sac.

Because après is a preposition that is followed directly by a noun phrase in French.

• après la réunion = after the meeting
• no extra de is needed

This is similar to other French prepositions that connect straight to a noun:

• avant le cours
• pendant le film
• après le déjeuner

So après la réunion is the normal structure.

The comma is there because Après la réunion is an introductory time phrase.

In English, we often do the same thing:

• After the meeting, I put the badge back in my bag.

In French, the comma is also common and helps readability, especially when the sentence begins with a time expression. It is not always absolutely required in short sentences, but it is very natural here.

Remets comes from the verb remettre.

This verb is built from:

• mettre = to put
• re-
  • mettre = to put back, put again, hand back, return

So je remets is the 1st person singular present form:

• je remets = I put back / I hand back / I return

The sentence uses remettre because the idea is not just put, but put back.

You could say je mets in some contexts, but it would lose the idea of putting something back.

Compare:

• je mets le badge dans mon sac = I put the badge in my bag
• je remets le badge dans mon sac = I put the badge back in my bag

So remettre adds the sense of returning something to its usual place or doing the action again.

French often uses the present tense to describe:

• a habitual action
• a routine
• a sequence of events
• something stated in a general, vivid, or narrative way

So je remets le badge dans mon sac can sound like:

• this is what I do after the meeting
• I’m describing my usual routine
• I’m narrating the action in a simple way

If you wanted a clearly future meaning, you could say:

• Après la réunion, je remettrai le badge dans mon sac.

If you wanted a past meaning, you could say:

• Après la réunion, j’ai remis le badge dans mon sac.

Le badge refers to a specific badge, not just any badge.

French uses the definite article when the speaker and listener both know which thing is being talked about, or when the object is understood from the situation.

So here:

• le badge = the badge, the known badge

If you said un badge, it would sound like an unspecified badge:

• je remets un badge dans mon sac = I put a badge into my bag

That is possible grammatically, but it changes the meaning.

Dans means in / inside. It is the normal preposition when something goes into an enclosed space.

So:

• dans mon sac = in my bag

It is used because the badge ends up inside the bag.

Other prepositions would change the meaning:

• sur mon sac = on my bag
• à côté de mon sac = next to my bag

So dans is the natural choice here.

Because sac is a masculine noun in French.

So the correct possessive is:

• mon sac = my bag

Compare:

• masculine singular: mon
• feminine singular: ma
• plural: mes

Examples:

• mon sac
• ma réunion
• mes affaires

So even though my does not change in English, French possessives must agree with the noun that follows.

Remettre follows the same pattern as mettre in the present tense.

Present tense:

• je remets
• tu remets
• il / elle remet
• nous remettons
• vous remettez
• ils / elles remettent

So je remets is simply the regular je form of this verb family.

A useful thing to notice is that the singular forms sound very similar:

• je remets
• tu remets
• il remet

The written endings differ, but the final -s in remets is not pronounced.

Yes, that is also correct.

Both are natural:

• Après la réunion, je remets le badge dans mon sac.
• Je remets le badge dans mon sac après la réunion.

The difference is mostly one of focus:

• Après la réunion, ... emphasizes the time first
• Je remets ... après la réunion starts with the action, then adds the time

French is fairly flexible with time expressions like this.

It is a borrowed word, but in French it is pronounced in a more French way, usually close to bahj.

A rough guide:

• badge → sounds like bahj
• the a is not the same as the English bad
• the final sound is like the j in measure or the s in vision

Also, in this sentence:

• le badge → the e in le is often very light in normal speech
• je remets le badge flows together smoothly, but there is no special mandatory liaison before badge

A rough English-friendly approximation is ray-oo-nyon, but the real French pronunciation is more compact and nasal at the end.

A few points:

• ré- sounds like ray
• u is the French u sound, which does not exist naturally in English
• -nion ends with a nasal sound, so the n is not pronounced like a full English n

Also, the accent in réunion matters because it shows the pronunciation of the first e:

• réunion, not reunion

And remember that French réunion does not sound like the English word reunion.

Yes. Remettre is a very common French verb with several meanings depending on context. Some common ones are:

• to put back
• to hand back
• to return
• to postpone
  
  • remettre une réunion = to postpone a meeting
• to give / hand over
• to recover in some expressions
  
  • s’en remettre has other meanings again

In je remets le badge dans mon sac, the context makes put back the natural meaning. That is a good example of how important context is with French verbs.