Why is the river Euphrates dried up?
The river Euphrates is dried up by the sixth angel's vial so that the way is prepared for the kings of the east to march their armies toward the final battle.
When the sixth angel pours out his vial upon the great river Euphrates, the water dries up so that the way of the kings of the east might be prepared. This is a direct act of God's judgment, clearing a path for a massive military movement from the eastern nations toward the land where the final conflict will take place.
Alongside the drying of the river, three unclean spirits like frogs come out of the mouths of the dragon (Satan), the beast (Antichrist), and the false prophet. These are spirits of devils who work miracles in order to deceive the kings of the earth and gather them together for battle. So the drying of the Euphrates is not just a natural event but part of a supernatural gathering of the world's armies to the place called Armageddon for the battle of that great day of God Almighty.
The drying of the Euphrates is echoed in Jeremiah, where the mighty men stumble and fall by the river Euphrates, and Egypt and other nations rise up like a flood to march north, only to be met with the sword of the Lord of hosts in a great day of vengeance. The removal of this great river barrier makes way for the massed armies of the kings of the east to advance toward the climactic battle that closes this stage of judgment.
The scriptures
And the sixth angel poured out his vial upon the great river Euphrates; and the water thereof was dried up, that the way of the kings of the east might be prepared. And I saw three unclean spirits like frogs come out of the mouth of the dragon, and out of the mouth of the beast, and out of the mouth of the false prophet. For they are the spirits of devils, working miracles, which go forth unto the kings of the earth and of the whole world, to gather them to the battle of that great day of God Almighty. Behold, I come as a thief. Blessed is he that watcheth, and keepeth his garments, lest he walk naked, and they see his shame. And he gathered them together into a place called in the Hebrew tongue Armageddon.
Let not the swift flee away, nor the mighty man escape; they shall stumble, and fall toward the north by the river Euphrates. Who is this that cometh up as a flood, whose waters are moved as the rivers? Egypt riseth up like a flood, and his waters are moved like the rivers; and he saith, I will go up, and will cover the earth; I will destroy the city and the inhabitants thereof. Come up, ye horses; and rage, ye chariots; and let the mighty men come forth; the Ethiopians and the Libyans, that handle the shield; and the Lydians, that handle and bend the bow. For this is the day of the Lord GOD of hosts, a day of vengeance, that he may avenge him of his adversaries: and the sword shall devour, and it shall be satiate and made drunk with their blood: for the Lord GOD of hosts hath a sacrifice in the north country by the river Euphrates.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.