The Bible itself answers the question:
What does "He that overcomes" mean?

By using an Interlinear Bible*, we can locate the Greek word that was translated as the English word "overcome" in the passage we are examining. Next, by using an Analytical Concordance, we can locate that specific Greek word in all other places in the Bible. By reading all the verses containing that Greek word, we can ultimately obtain a fuller, more accurate understanding of what the original authors were attempting to convey.

nikao Greek Strong's NR. 3528; to subdue (literal or figurative) :- conquer, overcome, prevail, get the victory.
In the whole NT this word is used in 28 places, whereof 15 in Revelation: (Luke 11: 22; John 16:33; Romans 3:4, 12:21; 1 John 2:13-14, 4:4, 5:4-5; Revelation 2:7, 2:11, 2:17, 2:26, 3:5, 3:12, 3:21, 5:5, 6:2, 11:7, 12:11, 13:7, 15:2, 17:14 en 21:7)

The word "overcome" clearly means that the enemy is overpowered and defeated in a victory that does not come easily, but rather through suffering or after a long or hard fight. The enemy is all that is evil. It cost Jesus His life. It will cost His followers tribulation.

John 16:33 (Jesus says) These things I have spoken unto you, that in me ye might have peace. In the world ye shall have tribulation: but be of good cheer; I have overcome the world.

The encouragement for ALL CHURCHES of ALL TIMES : Romans 12:21 Be not overcome of evil, but overcome evil with good.

The individuals who overcome will be the true victors with Christ. They will inherit and be called sons.
Rev. 21:7 "He who overcomes will inherit these things, and I will be his God and he will be My son."
The next verse talks about those who didn't overcome Rev. 21:8 "But for the cowardly and unbelieving and abominable and murderers and immoral persons and sorcerers and idolaters and all liars, their part will be in the lake that burns with fire and brimstone, which is the second death."

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* a Bible that contains, word for word, both the original Greek text and its English translation but does not bother with English grammar.