There is two ways that she could have come to this configuration from the previous time period:
(1) One case is that she would have moved a server from r to r' with the other server "parked" at x. Then the cost simply is w(x) + rr'.
(2)The other case is that she did not move the server at r to r', but rather served with the "other" server (i.e the one that did not serve the previous request.) We assume that this other server was at a point, say, y. Now the cost is min {w(y) + yr'} + rx. Note that w(y) + yr' achieves its minimum at y=r'. This means that min {w(y) + yr'} + rx = w(r') + rx.

Therefore, taking the minimum over case (1) and (2), we have w'(x)= min{w(x) + rr',w(r') + rx}