There are two assumptions you can make based on this chart. The first is still more or less true today: as demand for a product goes up, supply increases, and price goes down. If the price gets too high, demand falls.Er, no. I guess even smart people like Gates get simple economics wrong sometimes. It's understandable, though, I've always felt the P/Q axes should be switched, since people always speak about the price as the independent variable.
He's conflating the idea of pricing along the line with the demand line itself. Yes, at higher prices, less quantity is demanded and vice versa. Similarly, if you can sell the thing for a higher price, more people will be interested in and able to sell it, so the quantity available will increase, and vice versa.
However, "supply" and "demand" refer to the whole lines themselves, not points along the lines. Changing supply and demand means the lines shift. If "demand for a product goes up", the demand line shifts to the right, the equilibrium quantity increases, but so does the price, the opposite of what he's saying.
His misconception then leads to an even larger error in understanding with the next part:
The second assumption this chart makes is that the total cost of production increases as supply increases.
This is false. What it says is that with if the price you can get for a good is high there will be more suppliers. The canonical example is oil: when it sells for $20/barrel, only Saudi Arabia and places with it easily accessible will be able to profitably drill for it, and so the quantity supplied will be low. But at $120/barrel, you can now use fracking, deep water drilling, and other much more expensive processes, and so the quantity supplied will be higher.
Supply and demand is not about any individual producer producing more, and so it's not particularly useful in analyzing the behavior of a single firm. It's about the aggregate effect of multiple different parties, each with their own respective infrastructures and costs, profitably contributing to the supply (or not) at different price points.