As Jesus spoke these words, everybody knew what Jesus meant. In the Roman world, before a man died on a cross, he had to carry his cross (or at least the horizontal beam of the cross) to the place of execution.
i. When the Romans crucified a criminal, they didnt just hang them on a cross. They first hung a cross on him.
ii. Carrying a cross always led to death on a cross. No one carried a cross for fun. The first hearers of Jesus didnt need an explanation of the cross; they knew it was an unrelenting instrument of torture, death, and humiliation. If someone took up his cross, he never came back. It was a one-way journey.
iii. In the real-life crosses of the Roman world, no one took them up (suggesting a voluntary action). Instead, crosses were impressed upon people, quite apart from their willingness. Here Jesus said that those who follow Him must voluntarily take up their cross.
iv. This isnt to suggest that we can choose our way to die a living death as followers of Jesus; but as the unchosen circumstances come into life, we choose to bear them as a way to daily die for Jesus glory.
Denying yourself means to live as an others-centered person. Jesus was the only person to do this perfectly, but we are to follow in His steps.
Immediately after telling his disciples of his coming suffering and death, Jesus told them they had to be prepared for similar treatment. The disciples of Jesus are those who have given their lives to Jesus, and they will be obedient to their master even if it leads to hardship, persecution and death.
They will no longer rule their own lives, but will deny themselves personal desires in order to please Jesus. In sacrificing the life that puts self first, they will find the only true life.
On the other hand those who live for themselves may gain what they want in the present world, but they will lose the only life of lasting value, eternal life ( Matthew 16:24-27).
Sources : David Guzik