"'After you, another kingdom will rise, inferior to yours. Next, a third kingdom, one of bronze, will rule over the whole earth. Finally, there will be a fourth kingdom, strong as iron - for iron breaks and smashes everything - and as iron breaks things to pieces, so it will crush and break all the others.'" NIV translation
Daniel explains to the king that what he saw was a string of kingdoms each getting weaker than the last. The second kingdom (silver) was the Media-Persia empire which replaced the Babylonian. The third (bronze) represented the Greek empire and the fourth was the Roman empire. What is history to us today was prophecy to Daniel and Nebuchadnezzar.
The Romans were known for their legions of military forces which were the finest of the time and they are the iron in the feet of the statue. But, that is not all that the Roman Empire was known for and we see that in the clay that was mixed in with the iron.
"As the toes were partly iron and partly clay, so this kingdom will be partly strong and partly brittle. And just as you saw the iron mixed with baked clay, so the people will be a mixture and will not remain united, any more than iron mixes with clay." NIV translation
The Roman empire had military strength but they were weak because they grew to love pleasures above all other things. The people they conquered were never truly brought into the kingdom and so, although they were mixed, they were still somewhat separate.
"'In the time of those kings, the God of heaven will set up a kingdom that will never be destroyed, nor will it be left to another people. It will crush all those kingdoms and bring them to an end, but it will itself endure forever. This is the meaning of the vision of the rock cut out of a mountain, but not by human hands - a rock that broke the iron, the bronze, the clay, the silver and the gold to pieces. The great God has shown the king what will take place in the future. The dream is true and the interpretation is trustworthy.'" NIV translation
Daniel speaks of the kingdom of God and its appearing through Jesus Christ. This has happened in that we, as followers of Jesus Christ, have been made an eternal kingdom but it is yet to be the earthly kingdom. This will be fulfilled in the future as we see in Revelation 21. Daniel assures the king that the events of the dream will come to pass. Just as the rock described was from God, the interpretation of the dream could have only been from God and not men. The king would have realized it was true because he had been able to tell the king what the dream was and interpret it.