In Matthew 5:3 Jesus says Blessed are the poor in spirit,for their's is the kingdom of heaven. What is the meaning of poor in spirit and why will their's be the kingdom of heaven?

While the Bible tells us to grow in the Lord, and continue to nourish ourselves with the word of God, here we see Jesus telling that the poor in spirit are blessed, and the kingdom of God is theirs. (Matthew 5:3)

What it is referring to is a person who accepts the real condition of his/her spirituality. For example, if I think I am already mature and spiritually fit - it may lead me to have a sort of spiritual pride. I will start to think that I know enough of the Bible, or others are worse sinners than I am. This pride will keep me from growing spiritually since I'll stop depending on God.
But in reality, if we compare ourselves to Jesus, who is our standard, we'll find that we have a lot of faults and mistakes and that there is a long way to go. This will help me to daily depend on God no matter how long I have been a Christian. What Jesus means by a person who is poor in the spirit - is a person who accepts his weaknesses and shortcomings, depends on God daily and seeks to grow to be more like Jesus all the days of his life.

Only those who have such humility will be able to grow in their spiritual life will be part of the kingdom of God. They will be able to obey the command of Jesus to seek first the Kingdom of God.(Matthew 6:33) Since they seek this kingdom, they will receive this Kingdom. (Hebrews 12:28,29)

Hope that's clear. You can also read the link below for an added insight.