Hello,
You are running into an Excel limitation. A slicer that filters a normal Table uses AutoFilter, and AutoFilter hides entire worksheet rows. Because the right table sits on those same rows, those rows get hidden too, so the right table appears to vanish. There is no switch to make a Table slicer ignore other content on the same rows.
The clean fix that keeps everything side by side is to drive the left side with a PivotTable plus a slicer instead of filtering the Table itself. Pivot filtering affects only the PivotTable and does not hide worksheet rows, so the right table stays visible.
Do this. Click any cell in your left Table, choose Insert > PivotTable, choose Existing Worksheet and pick a cell on the left where you want it, then build the Pivot to show the fields you need. With the Pivot selected, go to PivotTable Analyze > Insert Slicer, pick the field you want to filter by, and place the slicer next to the Pivot. Clear any filters on the original Table, and stop using its slicer. Now filter with the Pivot slicer. The right static table will no longer be affected.
Let me know how it goes!