Here is what I understood from critical slowing down.
When we are near a phase transition, the autocorrelation time $\tau$ is very long.
Imagine we are doing MC simulations on a ferromagnetic near $T_c$ where $T_c$ is the temperature of phase transition. We are studying the observable $M$ which is the total magnetisation. The fact that $\tau$ is long physically mean that if I am in a microstate with magnetisation far from the average magnetization, it will take a long time (many steps) for the simulation to access to microstates that have magnetisation near the average magnetisation.
And, as the microstates that have magnetisation near the average magnetisation have a more statistical importance in the calculation, it means that we will have a very bad statistics if we don't sample long enough.
Am I right in this explanation?
Why is $\tau$ very long near a phase transition? I am not sure to get it.
Why are cluster algorithm better to treat such problems? This question has been asked here Critical slowing down in Monte-Carlo algorithm for classical 2D Ising Model but not really answered.