No way to know exactly unless they provide you with the code to read it but most chips change the fuel mapping by adjusting the injector timing/duration and/or fuel pump pressure if it's adjustable. If they are requiring 91 octane in place of 87, it probably means they're firing the injectors sooner/longer in a way that would make it more likely to detonate on 87 octane. I've never heard about higher octane fuel damaging an engine. 91 has no more energy than 87, it just has a higher auto ignition temperature and burns slower.
The chip advances ignition timing to allow the engine to fire earlier therefore causing a complete burn of the slower burning premium fuel. As premium fuel burns longer, it creates more power. Your stock ecu cannot adjust the ignition timing enough to get a complete burn using premium fuel. So YES, premium fuel does create more power because it burns slower. Just like magnum gunpowder, it also burns slower, creating more power.