In time series data, non-stationary data is first made stationary (Using Differencing or any other methods). We train the model using this stationary data. So how come model's forecast will be close to original non-stationary data (as model has been trained with stationary data)?
Stationarity is an issue mainly for ARIMA models. The "I" in ARIMA stands for "integrated".
To make a series stationary using the ARIMA method, you difference it .
After the model is estimated and forecasts are generated, the final step is to (re)-integrate the series to obtain forecasts that are close to the original data.
This applies for other methods besides ARIMA, for example Neural Networks or other ML methods: If you use some transformation to normalize or stationarize the data, then the final step in the forecasting process will be to reverse transform the data accordingly.