The context of the passage makes the meaning more clear.
Then said they unto him, Where is thy Father? Jesus answered, Ye
neither know me, nor my Father: if ye had known me, ye should have
known my Father also. (John 8:19, KJV)
These words spake Jesus in the treasury, as he taught in the temple:
and no man laid hands on him; for his hour was not yet come. (John
8:20, KJV)
Then said Jesus again unto them, I go my way, and ye shall seek me,
and shall die in your sins: whither I go, ye cannot come. (John 8:21,
KJV)
Then said the Jews, Will he kill himself? because he saith, Whither I
go, ye cannot come. (John 8:22, KJV)
And he said unto them, Ye are from beneath; I am from above: ye are of
this world; I am not of this world. (John 8:23, KJV)
I said therefore unto you, that ye shall die in your sins: for if ye
believe not that I am he, ye shall die in your sins. (John 8:24,
KJV)
Then said they unto him, Who art thou? And Jesus saith unto them, Even
the same that I said unto you from the beginning. (John 8:25, KJV)
I have many things to say and to judge of you: but he that sent me
is true; and I speak to the world those things which I have heard of
him. (John 8:26, KJV)
They understood not that he spake to them of the Father. (John 8:27, KJV)
Then said Jesus unto them, When ye have lifted up the Son of man, then
shall ye know that I am he, and that I do nothing of myself;
but as my Father hath taught me, I speak these things. (John 8:28,
KJV)
And he that sent me is with me: the Father hath not left me alone;
for I do always those things that please him. (John 8:29, KJV)
In just the portion of the chapter quoted above, the "that I am he" phrase occurs twice: once each in verse 24 and in verse 28.
Interestingly, the second occurrence of this phrase comes just after the verse that tells us the Jews had not understood that Jesus was speaking of His Father.
Relationship Between Jesus and the Father
The passage includes numerous references to Jesus' Father, the One who had sent him, and the one whom we should know if only we knew Jesus. Jesus makes clear that the things he is speaking he has received from the Father (vs. 28). What, then, do we understand to be the source of Jesus' words? Certainly the Father has perfect right to claim divinity, and if Jesus speaks His words, that divinity shines through.
About Italicized KJV Words
One does need to be careful of assuming that italicized words in the KJV do not actually belong, especially with anything in the New Testament (from Greek) because Greek grammar and language style often deliberately omits pronouns upon subsequent occurrences of them, leaving them as simply understood based on their default antecedent. (Greeks apparently did not like "unnecessary" repetition.) For those of us who like things to be less ambiguous, this fact can be a little disconcerting--but it does require a little more diligence to ascertain the connection within the Greek grammatical sentence structure.
So, as with Hebrew where "context is everything," with Greek, context is also paramount. Basically, on those italicized words, the ones that involve subjects, objects, or pronouns could be implied by the grammar. These implied words may be fully legitimate requirements to the sentence, and yet, not being explicitly spelled out in the text, might be italicized as additions in the KJV.
Why the Addition of "He"?
The grammar of the sentence makes little sense without this. It is very likely a case where the Greek has dropped the pronoun and it needs to be added into the translation.
In the Old Testament expression from which the "I AM" declaration, the expression is in better grammatical form.
And God said unto Moses, I AM THAT I AM: and he said, Thus shalt thou
say unto the children of Israel, I AM hath sent me unto you. (Exodus
3:14, KJV)
The full expression "I AM that I AM" makes clear that this declaration is definitive. The later use of "I AM hath sent me" (to be spoken by Moses) parallels what Jesus says in John 8 when he says the Father had sent him.
In John 8:24 & 28, the expression is "that I am"--grammatically different from the declaration made to Moses. If someone says "I am", with no particular context, it would sound a bit awkward, grammatically. It could mean that the person exists, but it would certainly not mean to claim divinity.
Conclusion
John seems to be trying to show wherein the Jews mistook Jesus' identity and why they rejected him. This is important for us so that we do not make the same mistake.