ANSWER: Jesus began to teach about the poor in spirit, mentioned in your question, near the beginning of his public ministry. He used this topic as the foundational teaching on which to explain and reveal the full intent of Old Testament laws. He also used it as the starting point on how true Christians should think and act in their everyday lives.
His discussion on the subject of the poor in spirit occurs during his famous Sermon on the Mount, where he states, "And He opened His mouth and taught them, saying, 'Blessed are the poor in spirit, for theirs is the kingdom of heaven . . .'" (Matthew 5:2 - 3, HBFV).
Surprisingly, only the book of Matthew (chapters 5 to 7) writes about the details of his message that opens with a discussion of the poor in spirit. The book of Luke (chapter 6) records only select pieces of his sermon. The opening sentences of his message are often called the Beatitudes or "beautiful attitudes" because they talk about the attitudes God highly prizes.
So, what exactly was the Lord discussing? One hint in understanding what Christ was referring to can be found by noticing the word "spirit" in his reference to the poor in Matthew 5:3. In the Holy Bible Faithful Version, and most other translations such as the KJV, NKJV, NIV and others, the word does not start with a capital letter. This signifies that the word is understood to not refer to the Holy Spirit, which is what makes someone a Christian (Romans 8:9), but to something else.
"Spirit" in Matthew 5:3 refers to a person's frame of mind or their attitude. It is how a person thinks about the world around them and their own personal relationship with the Eternal. Barnes' Notes on the New Testament, commenting on Jesus's teachings, states to have a poor spirit means to have a lowly or humble view of ourselves.
The Biblical Illustrator states that this poorness involves humility, contentment, submission and gratitude. A Commentary on the Holy Bible says that "Poverty of spirit is the opposite of pride, self-righteousness, and self-conceit . . ."
An excellent contrast between the self-exalting arrogance God dislikes (especially among religious people) versus the contrite heart he is looking for can be discovered in the parable of the Pharisee and the publican (Luke 18:10 - 11, 13 - 14).
Those who are poor in spirit realize, in the core of their being, their utter need for God and their inability to save themselves. They are keenly aware that they humbly need his help and mercy every moment of every day. Those who have this attitude are indeed blessed. Jesus stated they should rejoice and be filled with joy for great is their reward in heaven (Matthew 5:12)! It is a state of mind that all Christians should strive to attain.
The man who is poor in spirit is the man
who has realized that things mean nothing,
and that God means everything.
William Barclay (1907 - 1978)