How is PACF analysis output related to LSTM?
When used for time series, LSTM (or any Neural Network approach) fall under the category of auto-regressive models. The most general formulation of an auto-regressive model is: 

$Y_t = f(Y_{t-1}, Y_{t-2},...,Y_{t-n})$ with n the number of past lags to include in your model. 

When using a neural network for univariate time series modeling, n will determine the number of input neurons to use (It could be n, or could be n+k, with k additional causal features). 

The PACF will help you determine n. 

The key here is that neural networks are a generalization of AR(p) models.