Ne razumijem čemu se smiješ.

Here čemu means something like to what or, more naturally in English, what ... at.

So:

• čemu se smiješ? = What are you laughing at?

In this sentence, čemu does not mean why. It points to the thing that someone is laughing at, not the reason in a broad sense.

Because the verb smijati se normally goes with the dative for the target of laughter.

So Croatian says:

• smijati se nečemu = to laugh at something
• smijati se nekome = to laugh at someone

That is why you get:

• čemu se smiješ = what are you laughing at?

and not što se smiješ in standard Croatian.

Because zašto and čemu ask slightly different things.

• Zašto se smiješ? = Why are you laughing?
  This asks for the reason or cause.

• Čemu se smiješ? = What are you laughing at? / What do you find funny?
  This asks about the object or target of the laughter.

So:

• Ne razumijem zašto se smiješ. = I don’t understand why you’re laughing.
• Ne razumijem čemu se smiješ. = I don’t understand what you’re laughing at.

In English these can sometimes feel close, but Croatian keeps the distinction more clearly.

Because the verb is smijati se, not just smijati.

This se is part of the verb, and you should learn it together with the verb:

• smijati se = to laugh

It does not literally mean to laugh oneself in normal English terms. It is just how Croatian forms this verb.

A good habit is to memorize verbs with se as whole units:

• smijati se = to laugh
• bojati se = to be afraid
• nadati se = to hope

They look the same, but they come from different verbs:

• smiješ from smjeti = you may / you are allowed
• smiješ se from smijati se = you laugh / you are laughing

So:

• Smiješ ići. = You may go.
• Smiješ se. = You are laughing.

In your sentence, the presence of se makes it clear that this is from smijati se.

Because se is a clitic: a short unstressed word that normally goes in the second position of its clause.

In the clause:

• čemu se smiješ

the first element is čemu, so the clitic se comes right after it.

That is why standard Croatian prefers:

• čemu se smiješ

not:

• čemu smiješ se

This second-position behavior is very common with Croatian clitics such as se, sam, si, je, ga, mi, ti, mu and so on.

Here čemu is in the dative singular.

It is part of the pronoun što (what), which changes irregularly:

• što = nominative/accusative
• čega = genitive
• čemu = dative/locative
• čime = instrumental

So the sentence uses čemu because smijati se takes the dative:

• smijati se čemu
• smijati se nečemu
• smijati se tome
• smijati se čemu?

The change from što to če- in the other forms is just part of this pronoun’s pattern.

Because Croatian usually leaves out subject pronouns when they are already clear from the verb ending.

Here:

• smiješ = you laugh / you are laughing

The ending -eš already tells you the subject is you singular, so ti is unnecessary unless you want emphasis.

Compare:

• Ne razumijem čemu se smiješ. = neutral
• Ne razumijem čemu se ti smiješ. = adds emphasis, like I don’t understand what you are laughing at

Yes, it is natural and correct.

It sounds neutral in grammar, but in real life the tone can vary:

• calm: I don’t understand what you’re laughing at
• annoyed: I don’t get what you’re laughing at
• confrontational: depending on voice and context

So the sentence itself is fine; the emotional effect depends a lot on intonation and situation.

Yes. On its own, it is a normal question:

• Čemu se smiješ? = What are you laughing at?

Your full sentence simply embeds that question inside a larger sentence:

• Ne razumijem čemu se smiješ.

Literally, that is something like:

• I do not understand [what you are laughing at].

So the second part is basically an indirect question.

Yes, but it is not exactly the same.

• Ne razumijem čemu se smiješ.
  = I don’t understand what you’re laughing at.

• Ne razumijem što je smiješno.
  = I don’t understand what is funny.

The second version focuses more on what is funny, while the original focuses on what you are laughing at. In many situations they are close, but they are not identical.