I believe that those verses refer to what will happen when Christ returns in the end times, which I believe has already happened.
It says there will be a time of distress such as has not been made to occur since there came to be a nation until that time. We can see that in the world right now if we are looking. And during that time every one who is found written down in the book will escape. This refers to the Book of God that has already been written; God is all-knowing so God knows what everyone's fate will be. When Christ returns some people who were spiritually dead will awake to everlasting life, which is a state of the soul that is near to God.
This verse means exactly what it says: the unrighteous will be cut off from everlasting life and the righteous will have everlasting life. If you read the preceding verses you will understand who Jesus considered the righteous ones:
Matthew 25:35-45 For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in: Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me. Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed thee? or thirsty, and gave thee drink? When saw we thee a stranger, and took thee in? or naked, and clothed thee? Or when saw we thee sick, or in prison, and came unto thee? And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me. Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels: For I was an hungred, and ye gave me no meat: I was thirsty, and ye gave me no drink: I was a stranger, and ye took me not in: naked, and ye clothed me not: sick, and in prison, and ye visited me not. Then shall they also answer him, saying, Lord, when saw we thee an hungred, or athirst, or a stranger, or naked, or sick, or in prison, and did not minister unto thee? Then shall he answer them, saying, Verily I say unto you, Inasmuch as ye did it not to one of the least of these, ye did it not to me.
46 And these shall go away into everlasting punishment: but the righteous into life eternal.