Air is compressible, therefore an air tank can act as a pressure reservoir and reduce pressure fluctuations. As noted above, it is the pneumatic analog of an electrical capacitor. The diaphragm pump produces fluctuations that we want to smooth out.
From electrical theory we know that a capacitor alone cannot smooth out ripples: it must be combined with a resistor or inductor to produce a filter. The same is true in pneumatics. The air reservoir alone is not enough, but when combined with a resistance it can form a smoothing filter. In the hot air station the resistance is formed from the outlet air pipe and hot air nozzle. The resistance to air flow in this path, when combined with the air reservoir, produces the effective "RC" filter that is desired to smooth the air flow.