What they claimed
The apostles didn't claim to have had a feeling, or a vision, or a new idea. They claimed something far more specific. They said they had seen a dead man alive again.
"To whom also he shewed himself alive after his passion, by many proofs, for forty days appearing to them, and speaking of the kingdom of God."
Acts 1:3
Paul lists the witnesses: Peter, then the twelve, then more than five hundred at once, then James, then all the apostles (1 Corinthians 15:5 to 7).
Told to stop
The authorities in Jerusalem ordered them to stop talking about Jesus. Peter and John answered:
"If it be just in the sight of God, to hear you rather than God, judge ye. For we cannot but speak the things which we have seen and heard."
Acts 4:19 to 20
So they were flogged. And they walked out "rejoicing that they were accounted worthy to suffer reproach for the name of Jesus" (Acts 5:41).
One after another
Then it got worse.
James, the brother of John. The first of the twelve to die, around AD 44. "And he killed James, the brother of John, with the sword" (Acts 12:2).
Peter. Jesus had told him how he would die: "when thou shalt be old, thou shalt stretch forth thy hands, and another shall gird thee, and lead thee whither thou wouldst not. And this he said, signifying by what death he should glorify God" (John 21:18 to 19). Around AD 64, under the Emperor Nero, Peter was crucified in Rome. The early Church writer Origen records that he was crucified head downwards, at his own request (as reported by Eusebius, Church History 3.1).
Paul. He had once hunted Christians down. He met the risen Jesus on the road to Damascus and spent the rest of his life preaching him. "I have fought a good fight, I have finished my course, I have kept the faith," he wrote, waiting for execution (2 Timothy 4:7). He was beheaded in Rome (Eusebius, Church History 2.25).
James, the brother of Jesus. He didn't believe in his brother during his ministry: "For neither did his brethren believe in him" (John 7:5). Then, Paul says, the risen Jesus "was seen by James" (1 Corinthians 15:7). James became the leader of the church in Jerusalem. Around AD 62, he was stoned to death. That isn't only in Christian sources. The Jewish historian Josephus records it: the high priest "brought before them the brother of Jesus, who was called Christ, whose name was James... he delivered them to be stoned" (Antiquities 20.200).
Written down at the time
This isn't legend built up centuries later. Around AD 96, within living memory, Clement, a leader of the church in Rome, wrote about Peter and Paul:
"Peter, through unrighteous envy, endured not one or two, but numerous labours, and when he had at length suffered martyrdom, departed to the place of glory due to him."
Of Paul he wrote that he was "seven times thrown into captivity, compelled to flee, and stoned," and finally "suffered martyrdom under the prefects."
The rest
Church tradition records that the other apostles also died for their testimony, far from home. Andrew in Greece. Thomas in India. Others across the Roman world and beyond.
John was the one who lived to old age. Even he was exiled for it, "in the island, which is called Patmos, for the word of God, and for the testimony of Jesus" (Apocalypse 1:9).
Why this is a big deal
People die for things they believe all the time. That proves they believed it, not that it's true.
The apostles are different. They weren't dying for something they'd been taught. They were dying for something they said they had seen with their own eyes. If they had made it up, they would have known. And every one of them could have saved his life with one sentence: it wasn't true.
Not one of them said it.