Entschuldigung, ich habe dich missverstanden.

Because Entschuldigung is a noun. All nouns are capitalized in German.

In this sentence, it is being used as a fixed expression meaning something like sorry or excuse me, but grammatically it is still a noun, which is why it starts with a capital letter.

They are similar, but not always identical.

• Entschuldigung = Sorry / Excuse me
• Es tut mir leid = I am sorry

Entschuldigung is very common for quick apologies, especially in everyday speech.
Es tut mir leid can sound a bit more personal or sincere, especially in more serious situations.

In this sentence, Entschuldigung sounds very natural.

Because Entschuldigung is being used as a separate introductory expression before the main clause.

So the sentence is structured like this:

• Entschuldigung,
• ich habe dich missverstanden.

That comma helps show a pause, similar to English Sorry, I misunderstood you.

This is the Perfekt tense, which is very common in spoken German for talking about past events.

The structure is:

• ich habe = auxiliary verb
• missverstanden = past participle

So:

• ich habe dich missverstanden = I misunderstood you

In everyday spoken German, this is usually more natural than the simple past:

• ich missverstand dich — grammatically possible, but not normal in conversation

In a normal German main clause with the Perfekt, the conjugated auxiliary verb goes in the second position, and the past participle usually goes to the end.

So the pattern is:

• Ich
  • habe
    • rest of sentence + missverstanden

This is standard German word order:

• Ich habe dich missverstanden.
• Wir haben ihn gesehen.
• Sie hat das vergessen.

Because missverstehen has an inseparable prefix: miss-.

Many German verbs with inseparable prefixes do not add ge- in the past participle.

Compare:

• verstehen → verstanden
• missverstehen → missverstanden

So not:

• missgeverstanden ❌

This is the same idea as with other inseparable-prefix verbs.

Because dich is the accusative form of du, and here it is the direct object of the verb.

The verb missverstehen works like to misunderstand someone:

• ich missverstehe dich = I misunderstand you

So the object must be accusative:

• du = subject form
• dich = accusative object form
• dir = dative form

That is why:

• ich habe dich missverstanden ✅
• ich habe du missverstanden ❌
• ich habe dir missverstanden ❌

You would replace dich with Sie:

• Entschuldigung, ich habe Sie missverstanden.

This is the formal singular or plural you.

Other versions:

• dich = informal, one person
• euch = informal, more than one person
• Sie = formal one person or formal several people

So:

• Entschuldigung, ich habe euch missverstanden.
• Entschuldigung, ich habe Sie missverstanden.

Yes, absolutely. That is also very common and natural.

Both mean roughly the same thing:

• Ich habe dich missverstanden.
• Ich habe dich falsch verstanden.

A small nuance:

• missverstanden is a single verb meaning misunderstood
• falsch verstanden is more literally understood incorrectly

In everyday German, both are fine.

The base verb is missverstehen.

Its important forms are:

• infinitive: missverstehen
• present: ich missverstehe
• simple past: ich missverstand
• past participle: missverstanden

So this sentence uses:

• habe
  • missverstanden

which is the normal spoken past form.

Entschuldigung can definitely stand alone.

For example, if you just want to apologize quickly, you can simply say:

• Entschuldigung!

If you want to explain the reason, you can add the rest:

• Entschuldigung, ich habe dich missverstanden.

Both are natural.

The tricky part for many English speakers is the ch in dich.

It is the soft German ich-Laut, not like English ch in church and not like k.

A rough approximation is:

• dich ≈ dikh with a very soft hissy sound after the i

It is the same sound you hear in:

• ich
• mich
• nicht

So in the sentence:

• ich habe dich missverstanden

both ich and dich contain that same soft ch sound.

Yes, very natural.

A native speaker could easily say:

• Entschuldigung, ich habe dich missverstanden.

Depending on the situation, they might also say:

• Sorry, ich habe dich missverstanden.
• Entschuldige, ich habe dich missverstanden.
• Tut mir leid, ich habe dich missverstanden.

But your sentence is completely correct and idiomatic.