Το αντίθετο του «ζεστός» είναι «κρύος».

It follows the pattern:

Το αντίθετο του X είναι Y
= The opposite of X is Y

So here:

• Το αντίθετο = the opposite
• του ζεστός = of warm / of hot
• είναι = is
• κρύος = cold

It is a very useful pattern in Greek for talking about opposites.

Because αντίθετο here is being used as a noun: the opposite.

In Greek, nouns usually take a definite article more often than in English.
So:

• αντίθετο = opposite
• το αντίθετο = the opposite

Also, αντίθετο is neuter singular, so the article is το.

του here means of the and shows the genitive case.

So:

• το αντίθετο = the opposite
• του ... = of ...

That is why the sentence means:

The opposite of ... is ...

Greek often uses the genitive where English uses of.

Good question. Normally, if ζεστός were being used grammatically in the sentence, the genitive would be ζεστού.

Compare:

• nominative: ζεστός
• genitive: ζεστού

But here the word is being mentioned as a word, almost like a dictionary entry or label: warm / hot. When Greek quotes a word as a word, it is very common to keep it in its basic dictionary form, ζεστός.

So:

• του ζεστός = of the word ζεστός
• not necessarily of a masculine thing that is warm

You may also see more strictly inflected versions in some contexts, but keeping the quoted word in its basic form is very common in teaching and explanation.

Because Greek dictionaries usually give adjectives in the masculine singular nominative form.

So the basic forms are:

• ζεστός = warm / hot
• κρύος = cold

But adjectives change depending on gender and number:

For ζεστός:

• masculine: ζεστός
• feminine: ζεστή
• neuter: ζεστό

For κρύος:

• masculine: κρύος
• feminine: κρύα
• neuter: κρύο

In this sentence, the speaker is naming the words themselves, so the basic masculine dictionary forms are used.

It can mean either, depending on context.

Commonly:

• ζεστός καφές = hot coffee
• ζεστό δωμάτιο = warm room
• ζεστός καιρός = warm weather

So ζεστός covers a range from warm to hot, and its opposite is often κρύος = cold.

Because κρύος is being given as the answer word after είναι.

Greek often leaves out the article in this kind of equation when identifying or naming something:

• Το αντίθετο του ζεστός είναι κρύος.

You are essentially saying:

The opposite of the word warm is cold.

If you added an article, it could sound more like you are talking about a noun or a concept rather than just naming the opposite word.

Yes, absolutely. This is a very useful model.

For example:

• Το αντίθετο του μεγάλος είναι μικρός.
  The opposite of big is small.

• Το αντίθετο του εύκολος είναι δύσκολος.
  The opposite of easy is difficult.

• Το αντίθετο του γρήγορος είναι αργός.
  The opposite of fast is slow.

Again, when you are naming words, you will often see the basic dictionary form used.

A simple pronunciation guide:

• Το ≈ toh
• αντίθετο ≈ an-THEE-the-to
• του ≈ too
• ζεστός ≈ zeh-STOS
• είναι ≈ EE-ne
• κρύος ≈ KREE-os

A few notes:

• ζ sounds like the z in zoo
• θ sounds like th in think
• αι in είναι sounds like e
• υ in κρύος sounds like ee

The stress marks show you exactly where to stress the word:

• αντίθετο → stress on -ντί-
• ζεστός → stress on -στός
• κρύος → stress on κρύ-
• είναι → stress on εί-

Stress matters in Greek, so it is worth paying attention to the accent marks.

Yes, Greeks really do use this kind of sentence, especially in:

• teaching
• dictionaries
• classroom explanations
• word games
• basic definitions

It is a natural and useful way to talk about opposites. In everyday conversation, someone might also simply say:

• Ζεστός; Το αντίθετο είναι κρύος.
  Warm? The opposite is cold.

But the full sentence is completely normal and clear.