Why does slowly decaying ACF indicate that a time series is non-stationary?
The (simplified) definition of a stationary process is that the mean and variance of the process are constant over time. 

If the ACF is slowly decaying, that means future values of the series are correlated / heavily affected by past values. If past values of the series are high, the future values should also be high. If the value goes up, future values will also go up (assuming positive auto-correlation here). This means the mean will change over time, which means that the process is non-stationary.