Quasistatic process are almost always in equilibrium. We know that equilibrium implies zero entropy change. And zero entropy change implies that the process is reversible. So why quasistatic doesn't imply reversible?
Consider the case of a gas expanding slowly against a piston, which has friction.
The friction will dissipate some energy into heat, and this will happen no matter now slowly you move the piston; the dissipation is always going to be at least $\mu N \, \Delta x$. This dissipation increases the entropy, so quasistatic doesn't imply reversible.