What did Jesus mean when He said, "I do not know you, depart from Me"?

Let's examine the entire text to which you refer.
Then one said to Him, "Lord, are there few who are saved?"

And He said to them, "Strive to enter through the narrow gate, for many, I say to you, will seek to enter and will not be able.

"When once the Master of the house has risen up and shut the door, and you begin to stand outside and knock at the door, saying, 'Lord, Lord, open for us,' and He will answer and say to you, 'I do not know you, where you are from,' then you will begin to say, "We ate and drank in Your presence, and You taught in our streets.'

"But He will say, 'I tell you I do not know you, where you are from. Depart from Me, all you workers of iniquity.'"
(Luke 13:23-27 NKJ)
We know that this parable is dealing with salvation (eternal life, going to Heaven) because of the question asked of Jesus in verse twenty-three.

Jesus responds that there's only one way to be saved. Verse twenty-four explains that salvation comes by entering into the narrow gate. We know from other Scriptures that the narrow gate is referring to Jesus and believing on Him to be saved (Matt. 7:13-14; John 10:9-10, 14:6, 3:14-18).

Verses twenty-five through twenty-seven deals with the end of the world and judgment day. Only Jesus knows those who accept Him as Savior.

The people who say, "we ate and drank in Your presence, and You taught in our streets," are those who attended churches that talked about the Scriptures and the Lord, but they themselves never accepted Jesus as their personal Savior.

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