In this verse, Jesus hints at something that will happen a few days later. Some of the disciples, He told them, would not experience death before they experienced, or saw, the Kingdom of God. As we often do, they likely interpreted that to mean Jesus would soon establish His Kingdom on earth, defeat the Romans, and establish a Davidic-style reign. Knowing how we like to day dream, the disciples were probably no different. They likely envisioned palaces, regal robes, and princehoods (or at least exalted positions) for themselves. If they had only known, they would have been shocked by what Jesus would do and what was to come.
In just a few days, some of the disciples would go with Jesus to the mountain where they would "see" the Kingdom of God, as Jesus was transfigured before them and joined by Moses and Elijah. It was not at all what they thought He meant when He said they would see the Kingdom of God, but they never forgot those amazing mountain-top sights.
Perhaps you never do this, but some us have trouble with making assumptions about the words of God. I have to be particularly careful when I decide to "claim" a verse because I want to presume that the promise it contains will unfold the way I want it to unfold. I, like most of us, would like to have my way, and for it to be an easy, comfortable way. I have to constantly remind myself that God's way, no matter what it is, will always be better than mine. Here it is again, that business of Thy will and not my own. It runs throughout Scripture and is particularly pertinent here.
What the disciples likely expected from these words had little to do with how events would unfold. Instead of a temporal kingdom on earth, Jesus was about to establish an eternal Kingdom that would transcend time, transcend earth. Instead of lives of ease, the disciples would be hunted like criminals, experience hardship and loss, mostly die martyrs' deaths, and be instrumental in spreading the gospel around the world. The events that they didn't expect are the very ones that made it possible for us to have the good news of Jesus today. What a good thing the truth of Jesus, rather than the daydreams of the disciples, dictated the events that followed.
As you read and "claim" Scripture for your own, be careful to avoid "dreaming up" what the fulfillment of those promises will be. Leave the plan to God, confident that His way is best. Just like Jesus said, some of those disciples saw the Kingdom of God before they died. It may not look like what we expect, but He always keeps His word.