Why do these cars require premium fuel? Is it worth it?

Strictly speaking, DSMs do not require premium fuel.  However, the car was designed to accept higher-octane gasoline for a reason.

Octane is not a measure of the amount of energy in gasoline - rather, it is a measure of how well the fuel resists preignition, also known as detonation or 'knock'.  Detonation is very hard on the engine, and must be avoided.  The DSM cars incorporate a knock sensor specifically for the purpose of detecting knock.  Higher-octane gasoline resists knock better than lower octane gasoline.

During normal operation, the engine control unit (ECU)  adjusts the engine timing as far forward as it will go.  This provides maximum power, but also increases the possibility of knock.  This is especially true for turbocharged or supercharged engines, such as the 2.0L turbo 4-cylinder found in the upper model DSMs.  Note that the ECU hears knock all the time - the problem has to be persistent to generate a reaction from the computer.

If the ECU detects excessive, continuous knock via the knock sensor, the engine timing is retarded until the knock goes away.  This results in a power loss, but saves the engine from damage.  So, if you fill up with 87 octane gas, your ECU will be forced to retard the timing quite a bit, and you will not get the rated power from the engine.  Not only that, but the ECU may also be forced to limit turbo boost levels to accommodate the lesser quality gasoline, which again means a significant power loss.  And yes, the ECU does have the power to do that - see [["My boost is not strong enough.  Is there a fix?]]".

Once you switch back to premium, the ECU will eventually advance the timing back to original levels - after a while.

For full details on this and the role of the boost control solenoid (BCS) refer to the September 22nd edition of The Diagnostic Port, byTechnomotive.