I cannot recreate that problem. The folder stays open for me. There may be some specific variables involved. A screenshot could provide some valuable clues.
As a test, please try pressing F2 after highlighting the files. Does that initiate the rename or does it also cause the folder to close?
Also try doing the same multi-file rename using the Windows 10 style Explorer. You can open the old Explorer by first opening the Control Panel and then click the Up icon once or twice (depends on your starting Control Panel view). Once you've got the old Explorer open, navigate to the same folder and try the multi-file rename. Does that work?
P.S. If you are interested in looking at third-party bulk rename tools, the one I recommend is Bulk Rename Utility.