The voltage is dictated by the cell chemistry. So if the electrodes are Zinc and MnO2 (all common primary cells, include alkalines), you will always get the 1.5V. These materials then serve as the energy source for the cell, so their amount dictates how much energy you may get out of the cell before they are spent and so the cell flat. Similar way the larger the cell internal surface, the higher is the possible rate of the chemical reaction giving off the current, so higher the current the cell is capable of giving off, so higher the output power. It translates into the usable capacity and power output (however the capacity is not fixed, it strongly depends on the current draw). Because the R20 (aka "D") is larger than e.g. R3 (a.k.a. "AAA"), it can deliver way higher current (in the Amps range for an intermittent type of load for the R20, vs about 50..100mA for the R3). So when your device is designed for e.g. R6 (aka AA), it means its power consumption is below about 200mA. If you connect there the R20 cells, the thing will work perfectly, because the R20 would be loaded just by the 200mA, but are designed for >1A, so very light load for them, hence the very long life. But the drawback is, they will most likely not fit into the battery compartment - they are just physically too large...
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