Summary: As the people prepared to cross the Jordan River, God commanded Joshua to instruct the Kohanim carrying the Ark to enter the water. The upstream waters stopped and formed a column while the Kohanim were able to advance to the centre of the river before the rest of the nation crossed on dry land into Canaan. Once the nation had crossed, the Kohanim ascended with the Ark and the waters returned.
A deeper look: After the nation had crossed the Jordan River, God commands Joshua to instruct the Kohanim to ascend from the Jordan (Joshua 3:16-18). The simple meaning is that they came out of the Jordan River carrying the Ark before the waters returned. Yet the Gemara understands their ascent to be literal; more than they carried the Ark out, the Ark carried them (Sotah 35a).
The Ark is referred to primarily in chapters three and four as אֲרוֹן בְּרִית־הַשֶׁם– the Ark of the Covenant of God. This refers to God’s covenant with the Patriarchs that He will give the Land of Israel to their descendents which is the primary theme of these two chapters; the Jewish people are finally beginning their conquest of the Promised Land.
Yet on one occasion it is referred to as אֲרוֹן הָעֵדוּת– the Ark of Testimony (Joshua 4:16), referring to the Ark bearing witness to the close relationship between the Jewish people and God. The phrase אֲרוֹן הָעֵדוּתis mentioned on eleven other occasions in Tanach, either in relation to the פָּרוֹכֶת, the curtain which separated the Ark from the rest of the Tabernacle or regarding God communicating to Moses or Aharon.
The Midrash explains that the termאֲרוֹן הָעֵדוּתinforms us of God’s power and might. When God spoke to Moses and Aharon, He was like a king lowering himself to speak to his subjects (Pesikta Zutra on Exodus 25:22). The פָּרוֹכֶת represents both the division between the King and His people (Rashi on Exodus 26:31) promoting a healthy distance between leader and subject and yet teaches in this verse in Joshua that the role of a king is to carry his subjects, not the other way round. Kingship does not assign power to merely subjugate, but allocates authority to inspire, motivate and guide the people to a better place.