The Deity of Jesus Christ

WHY did the people take up stones to stone Jesus?

Because they understood exactly what He was saying, that he is God.

His claim, that HE is the I AM:
"Jesus said to them, 'Most assuredly, I say to you, before
Abraham was, I AM.' Then they took up stones to throw
at Him; but Jesus hid Himself and went out of the Temple,
going through the midst of them, and so ...passed by."
-John 8:58

What did He mean when he said, "I AM" ???

That he is the "I AM" as revealed to Moses:
"Then Moses said to God, 'Indeed, when I come to the children
of Israel and say to them, `The God of your fathers has sent me
to you,' and they say to me, `What is His name?' what shall I say
to them?'

And God said to Moses, 'I AM WHO I AM.' And He said, 'Thus
you shall say to the children of Israel, I AM has sent me to you.'
...The Lord God of your fathers, the God of Abraham,the God
of Isaac, and the God of Jacob, has sent me to you.

This is my Name forever, and this is My memorial to all
generations."
He also made this claim when questioned by the high priest:
"Again the high priest asked him, and said unto him,
Are you the Christ, the Son of the Blessed?
And Jesus said, I AM.
And you shall see the Son of man sitting on the right hand of power,
and coming in the clouds of heaven."
Mark 14:61-62
And again when speaking to a woman at a well in Samaria:
"The woman said to him,
'I know that Messiah is coming (who is called Christ).
When he comes, he will tell us all things.'
Jesus said to her, 'I who speak to you am he.'”
John 4:25-26