to cut it short the way the flasher unit works is that the power flowing through a bimetal strip inside it makes the strip bend, and this breaks the circuit, causing the indicator to switch off. As the metal strip cools, it straightens out again and makes contact, current goes through it, indicator lights, strip gets hot, it bends and repeat.
Anything that causes the amount of curent going through that strip to change (bad connection, burned bulb, or incorrect bulb) will alter the rate at which this process happens, and therefore the flashing rate. If for example you put a really powerful bulb in one of the indicators it would flash very slowly.
I'm pretty sure most modern cars don't use this any more, instead they probably use an electronic circuit since the clicking noise is now synthesised but they still flash faster when a bulb is out, this must be programmed into their function as people have become used to this behaviour to signal a burned bulb.