Why die rather than offer a pinch of incense? What is martyrdom for?

Ignatius of Antioch d. c. 110 ADLetter to the Romans 4.1–2

I am writing to all the churches and telling everyone that I die for God willingly, unless you hinder me. I beg you, do not show me untimely kindness. Let me be food for the beasts, through which I can reach God. I am God's wheat, and I am ground by the teeth of the beasts so that I may be found pure bread of Christ. ... Then I will truly be a disciple of Jesus Christ.

Martyrdom as imitation of Christ's passion and as Eucharist ("pure bread").

Pliny the Younger (Roman governor) c. 61–c. 113 ADLetter 10.96, to the emperor Trajan 10.96.3–5

I asked them in person whether they were Christians. Those who confessed I asked a second and third time, threatening punishment; those who persisted I ordered to be led away. For I did not doubt that, whatever it was they confessed, their stubbornness and unbending obstinacy deserved to be punished. ... Those who denied that they were or had been Christians, when they called on the gods after me, and offered incense and wine to your image, which I had ordered brought for this purpose with the statues of the gods, and moreover cursed Christ, none of which those who are really Christians can be forced to do, I thought should be released.

The test was a pinch of incense and wine before the emperor's image, and cursing Christ. The name itself was the crime.

Martyrdom of PolycarpMartyrdom of Polycarp 9.3

When the proconsul pressed him and said, "Swear, and I will release you; curse Christ," Polycarp said: "Eighty-six years I have served him, and he has done me no wrong. How can I blaspheme my King who saved me?"

"Martyr" means "witness". Denying Christ would be calling him a liar.

Martyrdom of PolycarpMartyrdom of Polycarp 4.1

One, named Quintus, a Phrygian lately come from Phrygia, lost his nerve when he saw the beasts. He was the one who had forced himself and some others to come forward of their own accord. The proconsul, pleading with him much, persuaded him to swear and to sacrifice. For this reason, brothers, we do not praise those who give themselves up, since the Gospel does not teach so.

Polycarp himself withdrew to a farm and waited to be found.

Tertullian c. 155–c. 220 ADApology 50.13

Yet none of your more refined cruelty does any good; it is rather an attraction to the sect. We grow more numerous every time you mow us down: the blood of Christians is seed.

The source of the saying "the blood of the martyrs is the seed of the Church".

Clement of Alexandria c. 150–c. 215 ADStromata (Miscellanies) 4.10

But if the man who kills a man of God sins against God, the man who brings himself before the tribunal is guilty of his killer's act as well. Such is the man who does not draw back from persecution, but hands himself over to be caught through rashness. As far as it rests with him, he becomes a partner in the persecutor's wickedness.

Clement is explaining the command to flee to another city (Matthew 10:23). Flight is permitted and even required; volunteering for arrest makes the Christian share the guilt of the killing, and real martyrdom for Clement is a whole life of witness, not a rushed death.

Origen of Alexandria c. 185–c. 254 ADExhortation to Martyrdom 22

For even if I escape for the present the punishment inflicted by men, I shall not escape the hands of the Almighty, either alive or dead.

Origen, writing to Ambrose and Protoctetus under arrest in the persecution of Maximinus, quotes the words of Eleazar in 2 Maccabees 6. Eleazar refused even a pretended eating of pork arranged to save him; Origen sets him and the Maccabean brothers before his friends as the pattern of a death chosen rather than a feigned denial.