All believers need to be able to explain why they believe in the resurrection. "He lives within my heart" or "I spoke to him this morning" may be fine as part of that, but it can't be the sum total of it. The bodily resurrection of Christ is a historical fact or Christianity is a joke.
The truth is that the resurrection of Christ is a story no one would make up. If you've never read that series before, please do. In all, I can combine that into one speech about 20 minutes long — and that is still leaving out some of the evidence.
Here I will boil it down into versions that can be shared in a few minutes or less.
(Bold is for subject headings/barest statements of facts. Normal text gives the thumbnail of the argument. Italics give explanatory comments that you can go into if time allows and if necessary.)
1. The death, burial, and resurrection of Jesus all happened publicly.
Jesus was executed in a public place and buried in a public cemetery. After his resurrection, his tomb was empty for all to see. Many of his post-resurrection appearances were in public places.
It is impossible to believe that Christianity could have begun in Jerusalem, within a short walk of Christ's tomb, unless that tomb was empty. The Jewish and Roman leaders would have been happy to produce Jesus' body to silence those who claimed he'd been raised from the dead (c.f., Acts 4). Instead, the oldest Jewish argument against the resurrection (the disciples stole the body) assumes an empty tomb.
2. The story contains many embarrassing elements that would have been omitted if untrue.
If first century Jews were to make up a story to start a new religion, it would not have included the founder being executed in what that society saw as the lowest, most degrading manner. They would not have women be the primary witnesses to the resurrection because that society viewed women as unreliable witnesses. They would not have made themselves look bad, being petty and argumentative and never understanding Jesus. And they wouldn't have included the fact that Jesus' own family didn't believe in him.
It's hard to grasp how that looked to people in their day. In modern terms, this story is as implausible as if the founder was put to death in the electric chair, the witness to the resurrection was the town drunk, his followers were childish high school dropouts, and Jesus' family had tried to have him committed to a mental hospital. It's not that these elements can't be true. It's that you wouldn't make these elements up if you were making things up.
3. There are many different appearance stories.
Jesus appeared to his followers under a variety of circumstances. It happened individually and in groups of varying sizes (from 2 to 500). He appeared in private rooms and in public places. He touched people and things. He even appeared to some who weren't his followers.
Some claim that the post-resurrection appearances were dreams, visions, or grief- (or drug-) induced hallucinations. But the variety of the appearances prevents that from being plausible. One person might convince himself that he saw Jesus, but not 500. They touched him. He cooked them lunch. Dreams don't cook.
4. There is no other explanation for the changes the resurrection caused.
Two changes that have to be explained are the changes in the disciples and the changes they made in their traditions. Cowards now boldly proclaimed that their crucified leader was Lord of the universe under threat of violence. And they proceeded to change the use and meaning of traditional ceremonies based on their belief that Jesus was the resurrected Lord.
First century Jews were very big on tradition. There was a right was to do everything, and it was the old way. But after Easter this group of Jews took the Passover meal and turned it into communion saying that there was a new covenant that did away with the centuries old sacrificial system. And these men, who ran and hid when Jesus was arrested, continued to teach this after being arrested and beaten and even when people started killing members of their group.
5. The story was prophesied long before it happened.
Dozens of elements of this unlikely story were foretold hundreds of years before it happened. Things no mere man could have controlled had to come together to fulfill these prophecies. Pick just eight prophecies — lots cast for His clothes, death by crucifixion, silent before accusers, badly beaten, killed with criminals, buried with rich, resurrection, betrayed for money. The odds of someone orchestrating those eight elements are 1 in a hundred million billion. Dozens of prophecies were fulfilled in the life, death, and resurrection of Christ.
One in a hundred million billion is a hard number to picture. A useful illustration goes like this: Cover the state of Texas two feet deep with silver dollars. Mark one of them and throw it out at random. Blindfold someone and let them walk as far as they want; then they pick up one silver dollar. The odds that they picked up the one you marked are the same as Christ fulfilling just eight prophecies.
These five elements of this story make for a story that you wouldn't, couldn't make up. That's why I believe it.
This argument requires that you believe that miracles are possible; that only requires that God exists. It also requires that you trust the Bible as a reliable source of information.
The above is drawn from many sources, chiefly The Case for Christ by Lee Strobel.