The fundamental point with the supply curve is that the marginal cost must be equal to marginal revenue. At that location, there is economic equilibrium because you will not be wasting revenue nor incurring extra marginal cost. Because of this, marginal cost and marginal revenue are intrinsically linked; therefore, the slope of the supply curve will move accordingly.
As supply increases, you will produce more. Initially, below equilibrium, the marginal cost will decrease with each piece produced. However, at the point where firms produce, the marginal cost increases for every product. At this point, marginal revenue increases as well, but slower. The point where the two cross is equilibrium, and it is at this point where you will want to produce.
The curve increases because, once you have produced a certain amount, it is not as valuable to continue to produce above that—so the marginal cost increases for each piece produced. To counter this, firms will raise prices if there is sufficient demand.