What is the real meaning of "The Father and I are one." (John 10.30) ?

Absolute, perfect unity between The Father and The Son. Jesus uses the word "are", indicating two persons. He further prays in ch. 17 of the gospel of John that His followers would be "one" as He and the Father are. Comparing these passages help us to understand the relationship between the Father and Son.

Those words enraged his fellow countrymen. They considered his statement to be blasphemous and were ready to stone him. (John 10:31-33)

But let's put this in context…a group of Jews had encircled Christ, demanding that he tell them clearly whether he was the Christ. Answering them, this is what he said: “I have told you, but you do not believe. The works I do in my Father's name are my witness; but you do not believe, because you are no sheep of mine. The sheep that belong to me listen to my voice; I know them and they follow me. I give them eternal life; they will never be lost and no one will ever steal them from me. The Father who gave them to me is greater than anyone, and no one can steal from the Father.

Clearly Jesus Christ was not claiming to be his Father's equal. He himself stated that he acted, not in his own name, but in the 'name of his Father.'