Mein Bruder sagt, der neue Stuhl kratzt auf dem Boden, wenn man ihn bewegt.

Because the rest of the sentence is a clause giving the content of what the brother says.

German uses a comma before:

• a subordinate clause, such as wenn man ihn bewegt
• and also before a content clause after verbs like sagen, denken, wissen, etc.

Here, Mein Bruder sagt, ... introduces what he says.

You could also say:

• Mein Bruder sagt, dass der neue Stuhl auf dem Boden kratzt, wenn man ihn bewegt.

In your sentence, dass is simply omitted.

German often leaves out dass after verbs like sagen, especially in everyday speech and writing.

So these are both possible:

• Mein Bruder sagt, der neue Stuhl kratzt auf dem Boden, wenn man ihn bewegt.
• Mein Bruder sagt, dass der neue Stuhl auf dem Boden kratzt, wenn man ihn bewegt.

There is a word-order difference:

• without dass: the next clause has normal main-clause word order
  → der neue Stuhl kratzt ...
• with dass: the verb goes to the end
  → dass der neue Stuhl ... kratzt

Because der neue Stuhl is the subject of the clause der neue Stuhl kratzt ...

The subject is in the nominative case.

• der neue Stuhl = nominative masculine
• den neuen Stuhl = accusative masculine

Here, the chair is the thing doing the action of kratzt, so it must be nominative.

Because the adjective neu comes after the definite article der and describes a masculine nominative singular noun: Stuhl.

The pattern is:

• der neue Stuhl
• den neuen Stuhl
• dem neuen Stuhl

So in der neue Stuhl, the adjective takes -e.

Here kratzt means something like scrapes or makes a scratching sound.

The verb kratzen literally means to scratch, but with furniture and floors it often means that something rubs against the surface and makes that unpleasant scraping noise.

So der neue Stuhl kratzt auf dem Boden means the chair scrapes against the floor when moved.

Because auf is a two-way preposition and can take either:

• dative for location
• accusative for direction/movement toward a place

Here, the idea is that the chair is scraping on the floor, meaning the floor is the surface where the scraping happens. That is a location, so German uses the dative:

• auf dem Boden

Compare:

• Der Stuhl steht auf dem Boden. = The chair is on the floor.
• Ich stelle den Stuhl auf den Boden. = I put the chair onto the floor.

It is dative because of auf used in a location sense.

Boden is masculine, and the dative singular form with the definite article is:

• dem Boden

So:

• auf dem Boden = on the floor

This is a very common pattern with two-way prepositions in German.

Because wenn man ihn bewegt is a subordinate clause, and German normally separates subordinate clauses with commas.

The word wenn means when or whenever, and it sends the verb to the end:

• wenn man ihn bewegt

So the comma is required.

Because wenn introduces a subordinate clause.

In German subordinate clauses, the conjugated verb usually goes to the end.

So:

• main clause: man bewegt ihn
• subordinate clause: wenn man ihn bewegt

That is one of the most important word-order rules in German.

Man means one, you, people, or someone in general.

It does not mean man as in adult male here.

So wenn man ihn bewegt means:

• when you move it
• when someone moves it
• when it is moved

English usually uses you or a passive construction instead of one in everyday speech.

Ihn refers back to den Stuhl.

Stuhl is a masculine noun, so its pronoun forms are:

• nominative: er
• accusative: ihn
• dative: ihm

In man ihn bewegt, the chair is the direct object of bewegt, so the accusative is needed:

• ihn

Yes. That would be a passive version:

• ..., wenn er bewegt wird.

That means when it is moved.

Your original sentence uses the more general active construction with man:

• wenn man ihn bewegt

Both are natural, but man is often very common in everyday German.

Because in a subordinate clause introduced by wenn, the conjugated verb goes to the end, while pronouns and other objects stay earlier in the clause.

So the order is:

• wenn
  • subject + object + verb
• wenn man ihn bewegt

That is normal German subordinate-clause word order.

Yes, absolutely.

That version is fully correct and means the same thing.

The main difference is structure:

• Mein Bruder sagt, der neue Stuhl kratzt ...
  = dass omitted, following clause keeps main-clause order
• Mein Bruder sagt, dass der neue Stuhl ... kratzt ...
  = explicit subordinate clause with dass, verb moves to the end

Learners should understand both, because both are very common.

Here it means when or whenever.

The idea is: each time the chair is moved, it scrapes on the floor.

German wenn can sometimes mean if, but in this sentence it is clearly about a repeated situation:

• wenn man ihn bewegt = when/whenever you move it

If the sentence were about a more hypothetical condition, English might translate wenn as if depending on context.