"It knows that if the spiritual life gets hold of it, all its self-centredness and self-will are going to be killed"

"Going to be killed" may sound like needlessly blunt language, but Lewis is taking the idea from Scripture.

Romans 8:13 (ESV): "If you live according to the flesh you will die, but if by the Spirit you put to death the deeds of the body, you will live."

Colossians 3:5-10 (ESV): "Put to death therefore what is earthly in you: sexual immorality, impurity, passion, evil desire, and covetousness, which is idolatry. On account of these the wrath of God is coming. In these you too once walked, when you were living in them. But now you must put them all away: anger, wrath, malice, slander, and obscene talk from your mouth. Do not lie to one another, seeing that you have put off the old self with its practices and have put on the new self, which is being renewed in knowledge after the image of its creator."