1 Corinthians 5 – Skeptic's Annotated Bible answered

1 It is reported commonly that there is fornication among you, and such fornication as is not so much as named among the Gentiles, that one should have his father’s wife.

5:1-5) "It is reported commonly that ..."
Paul, judging from rumors alone, complains that there are fornicators among his followers in Corinth; he is even worried that some have had sex with their fathers' wives. He says that those who have done these things should be "delivered to Satan for the destruction of the flesh" so that their soul can be saved.

2 And ye are puffed up, and have not rather mourned, that he that hath done this deed might be taken away from among you.

3 For I verily, as absent in body, but present in spirit, have judged already, as though I were present, concerning him that hath so done this deed,

(5:3, 12-13) "I ... have judged already, as though I were present."
To judge or not to judge

4 In the name of our Lord Jesus Christ, when ye are gathered together, and my spirit, with the power of our Lord Jesus Christ,

5 To deliver such an one unto Satan for the destruction of the flesh, that the spirit may be saved in the day of the Lord Jesus.

6 Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump?

7 Purge out therefore the old leaven, that ye may be a new lump, as ye are unleavened. For even Christ our passover is sacrificed for us:

8 Therefore let us keep the feast, not with old leaven, neither with the leaven of malice and wickedness; but with the unleavened bread of sincerity and truth.

9 I wrote unto you in an epistle not to company with fornicators:

"I wrote unto you in an epistle"
Despite its title, First Corinthians wasn't the first epistle that Paul wrote the Corinthians. There was another (Zeroth Corinthians?) that was either lost or considered not worthy of including in the Bible.

10 Yet not altogether with the fornicators of this world, or with the covetous, or extortioners, or with idolaters; for then must ye needs go out of the world.

11 But now I have written unto you not to keep company, if any man that is called a brother be a fornicator, or covetous, or an idolater, or a railer, or a drunkard, or an extortioner; with such an one no not to eat.

12 For what have I to do to judge them also that are without? do not ye judge them that are within?

13 But them that are without God judgeth. Therefore put away from among yourselves that wicked person.

(5:9-13) "Put away from among yourselves that wicked person."
Stay away from "fornicators", "idolaters", and "drunkards". Do not associate, speak to, or eat dinner with such "wicked" people.
Jehovah's Witnesses use these verses to justify disfellowshipping those who fail to follow the teachings of the Governing Body. JWs are told to shun disfellowshipped family and friends. They are not to speak to them or share a meal with them. If a witness is caught doing so, they will be disfellowshipped as well.