Sometimes this text has been used to keep oppressed people oppressed. "Take up [your] cross" has been quoted to keep women in abusive relationships, or the poor in poverty. Those are not the words of the Jesus that I know and love. My Jesus would never ask people to remain in dangerous or deplorable situations when they can get out. Jesus was speaking to people who were already bearing crosses - living oppressed and bound by an occupying government. His words were meant as encouragement, to give suffering people hope.
Look at the last line, "...before they see the kingdom of God." God's kingdom is this kingdom - our earth, our world. The prayer petition, "thy kingdom come, thy will be done on earth as it is in heaven," suggests that this world is where God's kingdom lives. When we act as though this world is God's kingdom, then this kingdom becomes the kind of world God intends for God's people.
So, when we hold onto hope in the midst of our suffering, the promise of God's kingdom becomes a reality. Our hope turns this world into God's kingdom. When we cling to God's words of hope and promise, then we see the kingdom of God.
Then he said to them all, “If any want to become my followers, let them deny themselves and take up their cross daily and follow me. For those who want to save their life will lose it, and those who lose their life for my sake will save it. What does it profit them if they gain the whole world, but lose or forfeit themselves? Those who are ashamed of me and of my words, of them the Son of Man will be ashamed when he comes in his glory and the glory of the Father and of the holy angels. But truly I tell you, there are some standing here who will not taste death before they see the kingdom of God.”