Setting up a physical monument serves as an everlasting witness to an open miracle. By taking stones directly from the riverbed, a tangible link is forged between the generation that actually walked through the split waters and all those who will come after them.
The initial instruction to speak about the event is directed at the people standing there right after the crossing. Since they just lived through the experience, the goal is not to inform them of what happened, but to clarify exactly why it happened. They needed to understand that the waters stopped flowing strictly because of the Ark of the Covenant and God's power, rather than any special merit of the people. This truth is proven by the exact timing of the event, as the river split at the precise moment the Ark passed through [מלבי״ם].
The miracle itself unfolded in two distinct stages, affecting different layers of the river. First, as the Ark merely approached the river and before it even entered, the upper layer of clear, drinkable water was cut off. Then, as the Ark moved directly through the riverbed, the murky, muddy waters sitting at the very bottom were also stopped [חומת אנך].
While the immediate explanation was for those who witnessed the river stop, the physical stones themselves were meant for the future. Placed in Gilgal, the stones stand as a permanent memorial for the generations who did not see the miracle with their own eyes [מלבי״ם], serving forever as a clear, undeniable symbol of the event [רש״י, ביאור שטיינזלץ].