Power required is drag times speed. In essence, you ask why drag at the same lift and in 10,000 ft is growing either side of the best climb speed at sea level.
As @Kolom rightly points out, the coincidence of best climb speed at sea level being equal to lowest drag speed at 10,000 ft is just that, a coincidence. There is no causal relation.
The power required curve is shifted right due to lower density at 10,000 ft which requires the aircraft to fly faster for the same dynamic pressure. The X-axis of the plot is True Airspeed, so all polar points will shift right as altitude increases. Next, you will also note a small upward shift. This is caused by the lower Reynolds number at higher altitude which increases zero-lift drag at the same indicated speed.
The best rate of climb speed is where the local slopes of the power required and the power available curves match. Then the distance between both power curves is greatest and leaves the highest specific excess power after the power required for steady flight has been subtracted from the available power. If the curves would be plotted over Indicated Airspeed, they would simply shift slightly up for the power required and down for the power available as density drops with increasing altitude. The author of your book chose to plot them over True Airspeed, so it might seem that the best ROC at sea level has some significance for the drag at 10,000 ft. It hasn't.
Please make sure you read the linked answers, too, because they should help to give more background. I have kept the answer short, so some things are not fully explained. For example, why there is a minimum drag point at all is explained at the target of the first link.