I can't understand this sentence: "The distribution theory underlying the use of the sample ACF and PACF as approximations of those of the true DGP assumes that the $y_t$ sequence is stationary" (from: Enders, Applied Econometric Time Series"
If your time series is nonstationary, then generally (!) there are no population counterparts of the sample ACF and PACF. Thus the sample ACF and PACF cannot converge to the population ACF and PACF. While you normally use the sample to make inference about the population or forecast a new sample from the population, nonstationarity does not allow to do that, because in general (!) there is no well-defined population to speak of (the population is changing, possibly unpredictably, with each new time point).
You have your time series $(y_t)$. That time series has certain ACF and PACF. You don't know how the random variables that make up your time series look like, so you can't calculate the ACF and PACF from them. You do know however, some data sampled from those random variables. From that sample you can calculate the sample ACF and sample PACF. There are results from distribution theory that tell you that the sample ACF and sample PACF will be an approximation for the ACF and PACF of the time series. These results are valid under the assumption that the time series has the property of being stationary.
Be aware that there are varying definitions of stationarity. You will have to figure out which one the author is referring to.
You look at ACF and PACF to get an idea of the lag structure of the process. For instance take a look at this plot: You see how ACF is declining in amplitude exponentially, while PACF cuts off after lag 1. This may suggest that you're dealing with AR(1) process.
How do I know this? If you derive the ACF and PACF assuming that your process is AR(1), which includes constant variance of errors, then you'd come up with a similar shaped curves. Hence, if your underlying series are not stationary, you're breaking the assumptions that are base for the heuristics that I mentioned about ACF/PACF. It's pointless to apply these on non-stationary series, since you can't make any conclusions about the lag structure anymore.