I'm not sure where you get that. Strong's lexicon gives the first definition as breath, and the word is translated in the Authorized Version, in the 24 places it appears, as "breath" 17 times, "blast" 3 times (as in a blast of breath -- 2 Samuel 22:16, Psalm 18:15), "spirit" only 2 times, "inspiration" 1 time, and "souls" 1 time.
Notice the same words, "BREATH OF LIFE" as used in Genesis 2:17 applied to animals in Genesis 7:21-22. It's BREATH. Air in the nostrils and lungs. The same word used in Deuteronomy 20:16, Joshua 10:40, Joshua 11:11, Joshua 11:14, 2 Samuel 22:16, 1 Kings 15:29, 1 Kings 17:17, Job 27:3, Job 33:4, Job 34:14, Psalm 150:6, Isaiah 2:22, Isaiah 42:5, Daniel 10:17. It's BREATH in all those places. In the majority of the places the word appears it is air in the nostrils and lungs. In only a very few places is that word used to signify anything other than air -- breath -- in the body. And "inspiration" is the opposite of "expiration," and "inspire" means to breathe, cause to breathe, inhale, while "expire" means to breathe out, exhale, emit one's last breath, die.
That the word would be used interchangeably with "souls" or "spirit" in those few other places is no surprise. A breathing man is a living soul (Genesis 2:17). And since man is a trinity (spirit, soul and body) a breathing man who is a living soul has a spirit. (Whether it is a dead spirit inherited from Adam, or a living one brought up by the quickening Spirit, depends on whether or not a man has been born again.)
And that brings us to John 3:5. The "water" birth is the physical birth -- the point when breath enters the body and a man becomes a living soul. It is one of the two requirements for a man to enter the kingdom of God. (Ever hear an expectant mother say "my water broke"? That's the water.) ("Water" also happens to be a scriptural type of the word of God.)
Obviously a man has to be born before he can be born again. So you first have a man who is a living soul because he has breath in his body. But he has a dead spirit that needs something. And is needed is the Spirit of God in him to quicken his dead spirit. [And the word "Spirit" in John 3:5 (if you want to run to the Greek) is Pneuma, from which we get "pneumatic" -- which we use today to indicate something driven by air.] So a man becomes a living soul when physical breath enters his body, and he gets a living spirit when the living God, like "breath" -- in the person of the Holy Spirit -- enters and quickens him. At that point the man is "born again." (See John chapter 3.)
And how is this accomplished? Very simply. By asking. By believing on Jesus Christ and simply inviting Him into your heart to be your Saviour.