I am not a specialist in the history of mathematics, but according to the Wikipedia article about Rhind papyrus (sections Volumes and Areas), the idea is as follows.
We take a circle (blue) and its circumscribing square, divide each side of the square in three equal parts and remove the corners. The resulting octagon (red) approximates the circle. If the side of the square is 1, then the area of the octagon is . So, for a circle of some diameter d, the corresponding octagon's area is . However, for some reason the Egyptian mathematicians wanted to express this as for some number . I am guessing that since is irrational, they decided to approximate it as . Indeed, . That's how they got the formula for the area of the circle.
Note that the octagon's approximation of the circle is not too bad because it both adds to and subtracts some areas from the circle. However, according to Wikipedia, "That this octagonal figure, whose area is easily calculated, so accurately approximates the area of the circle is just plain good luck".
If you need a more reliable source than Wikipedia, you should probably look for a book about the history of or ancient Egyptian mathematics.