The next two verses are helpful:
27 But I tell you truly, there are some standing here who will not taste death until they see the kingdom of God.”
28 Now about eight days after these sayings he took with him Peter and John and James and went up on the mountain to pray. 29 And as he was praying, the appearance of his face was altered, and his clothing became dazzling white. 30 And behold, two men were talking with him, Moses and Elijah, 31 who appeared in glory and spoke of his departure, which he was about to accomplish at Jerusalem. (Luke 9:27-31)
Possibly, Jesus was referring to this “transfiguration” event. At that time, Peter, John and James saw Jesus in glory, probably as he would appear in the kingdom. So in this way, the “saw the kingdom of God”. It is even possible that they were transported in time to the kingdom (hence the presence of Moses and Elijah) and so were seeing the kingdom in a literal sense as well.
Peter refers to the transfiguration in his second letter:
For we did not follow cleverly devised myths when we made known to you the power and coming of our Lord Jesus Christ, but we were eyewitnesses of his majesty. 17 For when he received honor and glory from God the Father, and the voice was borne to him by the Majestic Glory, “This is my beloved Son, with whom I am well pleased,” 18 we ourselves heard this very voice borne from heaven, for we were with him on the holy mountain. (2 Peter 1:16-18)
Notice that he links the transfiguration with the “power and coming of our Lord Jesus Christ”, lending support to the above interpretation.