Why are demand and supply curves shown as concave up? Why is it that demand and supply curves are typically shown in graphs, including in Econ 101, as concave up (i.e., convex)? 

I know what it entails mathematically—that if $p(q_d)$ is a demand curve, $p''(q_d)$ is positive, and if $p(q_s)$ is a supply curve, $p''(q_s)$ is positive, but what’s the intuition for why that’s usually the case?
This is so because usually the marginal costs of firms are increasing in production and many economists think also at increasing rate.

Intuitively you are only willing to provide good Q if the marginal costs are less or equal to price $MC \leq P$. Providing goods to the market at price below marginal cost is irrational and hence inconsistent with basic assumptions behind supply and demand.

The reason why marginal costs are thought of as usually increasing is that the more you use up some factor of production let’s say capital the scarcer it becomes and hence the return to capital increases and hence the MC increases. Moreover, many economists believe that they also do this at increasing rate (That is if you increase the consumption of labor twice the labor costs will increase more than twice) and hence you get convexity. But this being said you can find concave cost functions and also linear cost functions as well which are neither convex or concave in many textbooks as well.