For engine fire, the reason to close the packs (affected side) is to prevent smoke from filling up the cabin? But why the Isolation Valve and APU Bleed Air need to be closed too?
Closing the isolation & APU bleed valves keeps hot bleed air away from areas near the engine that may have damaged bleed air ducting. Closing the Pack valve causes the other pack to switch to high flow, making up for some of the lost output of the affected-side's pack. The steps are together in the checklist because it's all on the same overhead panel, even though the reasons differ.
I don't fly B737 aircraft. I would imagine the APU bleed air is used to start the engine. The other way is to probably use a huffer cart.
If there is an engine fire, that smoke can travel the same path back to the APU and then that smoke can be brought into the cabin through the other pack. So the idea, is to prevent any smoke from entering the cabin either directly from the engine or indirectly through the APU.