There is a little bit of confusion here.
The AR(p) is weakly stationary by definition:
with $|\phi_1| \le 1$.
Under these hypothesis you can prove that both mean, variance and autocovariance do not depend on time, hence it is a weakly stationary process.
So that was the theoretical part. Now there is the model estimation. If you fit an AR(p) to your data and that is the true data generating process, you should find (statistical evidence) that residuals are not autocorrelated (as well as rejecting hypothesis of unit root). If that's the case then you can use that model for prediction.
In my opinion the first sentence of your question is referred to this aspect.
Stationarity DOES NOT imply absence of autocorrelation
In fact AR(1) is a stationary but autocorrelated process. In the theoretical model the errors are $i.i.d.$
The random walk process is not stationary despite having $i.i.d$ errors.
see Autocorrelation vs Non-stationary