Je veux juste me reposer ce soir.

Because se reposer is a reflexive verb in French. It means to rest / to relax oneself.

In the infinitive, the full verb is se reposer.
When it follows je veux, the reflexive pronoun changes to match the subject:

• je veux me reposer = I want to rest
• tu veux te reposer = you want to rest
• il veut se reposer = he wants to rest

So me is there because the speaker is I / je.

In French, when one verb is followed by another in the infinitive, the reflexive pronoun usually stays with the infinitive verb, not with the conjugated verb.

So the structure is:

• je veux me reposer
• literally: I want myself to rest

This is normal in French with verbs like:

• je vais me coucher = I am going to go to bed
• je peux me lever = I can get up
• je dois me dépêcher = I must hurry

Putting me before veux would sound wrong here.

Here juste means just, only, or simply.

So Je veux juste me reposer ce soir means something like:

• I just want to rest tonight
• I only want to relax tonight
• All I want is to rest tonight

It often gives the idea that the speaker does not want anything complicated or demanding.

French adverbs like juste are often placed before the infinitive they modify.

So:

• Je veux juste me reposer
  means I just want to rest

The juste is connected to the idea of me reposer, not really to veux alone.

You may also hear slightly different placements in spoken French, but Je veux juste me reposer is very natural and common.

Ce soir can mean either this evening or tonight, depending on context.

Examples:

• Je sors ce soir = I’m going out tonight / this evening
• Je veux me reposer ce soir = I want to rest tonight

French uses ce soir where English may choose either expression.

Because soir is a masculine noun in French: le soir.

So the correct demonstrative adjective is:

• ce soir = this evening / tonight

Compare:

• ce matin = this morning
• ce jour = this day
• cette nuit = tonight / this night

You use cette only with feminine singular nouns.

Yes. Seulement can also mean only / just.

So this is correct:

• Je veux seulement me reposer ce soir

But there is a slight difference in feel:

• juste is very common and natural in everyday speech
• seulement can sound a little more formal, careful, or emphatic depending on context

In casual spoken French, juste is extremely common.

Not exactly, though they are similar.

• se reposer = to rest, to get some rest
• se détendre = to relax, to unwind

In many situations, both can work, but the nuance is different:

• Je veux me reposer ce soir suggests being tired and wanting rest
• Je veux me détendre ce soir suggests wanting to relax and unwind

Sometimes English relax could be translated by either one depending on the situation.

Because French does not use a separate word like English to in this structure.

In English:

• I want to rest

In French:

• Je veux me reposer

The infinitive form itself already carries the idea of to + verb.
So reposer or se reposer is the infinitive, and no extra word is needed.

It would usually be pronounced something like:

• Je veux juste me reposer ce soir
• approximately: zhuh vuh zhoost muh ruh-poh-zay suh swar

A few useful pronunciation notes:

• je sounds like zhuh
• veux sounds roughly like vuh
• juste has the zh sound at the start
• me reposer flows together smoothly in speech
• ce soir sounds roughly like suh swar

In fast speech, French rhythm links words together smoothly, so the sentence may sound more connected than an English speaker expects.

You can roughly map it word by word:

• Je = I
• veux = want
• juste = just
• me reposer = rest / relax
• ce soir = tonight / this evening

But a fully natural English translation is better than a strict word-for-word one. Depending on context, natural English could be:

• I just want to rest tonight.
• I just want to relax tonight.
• I only want to take it easy tonight.

So the sentence structure is similar to English, but it should still be understood as a normal French expression rather than translated mechanically every time.