Ο μάγειρας είπε ότι το φαγητό θέλει λίγο ακόμα πιπέρι, αλλά καθόλου άλλο αλάτι.

It introduces a reported statement after είπε.

• είπε ότι ... = said that ...
• It works very much like English that in reported speech.

So:

• Ο μάγειρας είπε ότι ... = The cook said that ...

A very common alternative in everyday Greek is πως:

• Ο μάγειρας είπε πως ...

Both are natural here.

What you would not use here is να, because είπε να... usually changes the meaning to something like said to... / suggested to... / told someone to...

No. They are different words.

• ότι = that
• ό,τι = whatever / anything that

In your sentence, it is ότι, because it introduces what the cook said.

Compare:

• Είπε ότι το φαγητό θέλει πιπέρι. = He said that the food needs pepper.
• Βάλε ό,τι θέλεις. = Put whatever you want.

The comma in ό,τι is there to distinguish it in writing.