It's caused by the fact that a double negative is a positive... no, really.
It is done with what is called feedback. The output of a circuit is used to set its own input. This means that when the original input goes away, the output is still keeping itself in the same state.
Changing the output generally involves overriding this 'feedback' input with an outside input.
As an example, inverters are logic gates that, when given an input of either LOW or HIGH, output the opposite, HIGH or LOW, respectively. Now, imagine two inverter gates, A and B, which are connected so that the output of A is the input of B, and the output of B is the input of A. Now, assume that for whatever reason, the input of A is high. Then the output of A will be LOW, causing the input to B to be LOW, causing the output of B to be high, and thus making the input to A HIGH, like we assumed. A double negative makes a positive. Thus, the circuit is in a stable state, A LOW, B HIGH. Now, suppose we somehow force the input of A to suddenly be LOW without changing anything else (perhaps by grounding the input with a pushbutton). Then the output of A goes high, the output of B goes low, and so it is again in a stable state, this time A HIGH, B LOW. Bill's transistor flipflop is really just this circuit; the two transistors are each configured as an inverter, where if the input at the base is a high voltage, the collector of the transistor is a low voltage.
Digital flip-flops use this situation to 'remember' that things have occured. There are lots of different kinds of flipflops, but they all use something like this as their basis. The differences have to do with how they get set to the different states. RS flipflops have two different inputs, one which sets it one way, the other of which sets it the other way. JK flipflops also do this, but use a clock signal to synchronize the input; the state only changes when the clock changes. T flipflops change state whenever a pulse is seen on the input. D flipflops are clocked, and set up so the output follows whatever is on the input when the clock transitions.
You can read this site to find out more: