Why do some SMPS power supplies require an input voltage select switch?
Most likely the switch controls how the mains input is rectified into the input bulk capacitors. This was common on older ATX power supplies as well, to use single design that can be used globally. Today the PFC stage at the input can handle larger input range automatically.

In the supply, there might be two 200V bulk capacitors for storing rectified mains. In the 230V position, the bridge rectifier is connected as a full bridge to the two series connected capacitors to effectively use them as a single 400V capacitor. Stored voltage is about 325V peak, or about 162V per capacitor. In the 115V position, the bridge rectifier is connected as a half bridge to a single capacitor, where positive mains cycle rectifies the 162V peak into one capacitor, and negative mains cycle rectifies the 162V peak into other capacitor. The total voltage over both capacitors would then be again 325V.

So if you use the switch in 230V position and feed in 115V, the input voltage is half what is needed and due to undervoltage it may not work at all or it might be so simple design that it just blindly tries to work but output halved as well. As this is not intended operation it may damage the parts.

Using it the other way, switch in 115V position but feeding in 230V will be much more dangerous and spectacular. The fuse might blow immediately, and if it does not, there will be a lot of overvoltage at the capacitors so they can heat up and vent up electrolytes as vapour after a while (seen this) or the capacitors may just explode outright.