The turbo blanket isolates the heat produced by your turbocharger, and prevents that heat from damaging, or even igniting, components surrounding the turbocharger within your engine compartment, such as plastic and rubber hoses and electrical wiring, as well as painted surfaces, such as the engine bay and the surface of the hood. Also, it prevents areas of localized high temperature from damaging the engine itself.
In keeping the exhaust gases within the turbocharger hot, turbocharger efficiency is improved. As you may know, the hotter a gas is, the more expansive it is. Within a contained system of a specified size, the more expansive a gas is, the greater the pressure derived and thus, the greater the flow of gas to escape the containment. With this increased pressure and flow rate for a given engine RPM, the acceleration of the turbocharger's impeller is increased as compared to the same turbocharger with the engine at the same RPM but with cooler exhaust gases. This equates to faster spool up of the turbocharger, as well as greater attainable levels of boost. What a driver will experience with a turbo blanket is greater turbocharger responsiveness. The faster spool up of the turbocharger means less turbo lag and a more linear power curve.
As you may know, it is very important to keep engine intake air cool. This is why intercoolers are often utilized with turbochargers. Similar to above, the cooler a gas is (such as intake air), the more dense it is. The more dense the intake air, the more oxygen it contains per unit volume. The more oxygen reaches the engine, the more power can be obtained. In keeping the heat of the exhaust gases contained within the hot side of the turbocharger and away from the cool side of the turbocharger and the intake path, more oxygen per unit volume reaches the engine, and thus, more power.