An Isolation transformer is a method for providing safety of life, in areas which are high risk for electric shock.
The way mains power is supplied is that the power company connects "neutral" to ground, and supplies the "pressure" (voltage) via the live wire. The safety risk here is that if you come into contact with the live wire, and the ground simultaneously, the current can flow through you to complete the circuit.
GFCIs (also called RCDs) detect when current in live and neutral don't match (the assumption being that the live wire must be coming into contact with ground), and shut off the power when this condition is detected. E.g. if I'm running an electric saw, cut through the cable, and pick up the end of the exposed live wire. The GCFI will trigger and shut off the power before the shock has time to cause significant health effects.
Isolation transformers are an alternative method. The transformer creates a new pair of live and neutral wires with no connection to ground (i.e. they are isolated) - the energy is transferred magnetically. If either of those wire comes into contact with ground, no current will flow, as the entire circuit is isolated. In the saw scenario above, if I picked up the live wire, I will receive no shock, as there is no circuit formed through me and the ground.
Isolation transformers have the advantage of simplicity, reliability, the passive nature of the safety (a GCFI must detect an unsafe condition and activate to ensure safety, a transformer is inherently safe) and robustness. So, they are preferable to GCFIs for certain activities - e.g. construction sites (where many countries make isolation transformers a legal requirement for power tools).
Isolation transfers are sometimes also used in hospitals, and other areas where very high power reliability is needed. Metal cased appliances are protected against electric shock, by connecting the chassis to ground. If the live wire comes loose and contacts the chassis, a short circuit will occur causing a fuse to blow, breaker to trip, or GCFI to trip out. In some circumstances, e.g. in a hospital OR, this is unacceptable. By using an isolation transformer, if a live wire contacts a metal chassis (or there is dust/water ingress, etc.), a short circuit will not occur, and power will remain on. Electric shock is prevented because the transformer provides isolation. This ensures that the faulty equipment can be removed, while ensuring that other equipment on the same circuit is not interrupted. Note that in this medical environment, sophisticated monitoring equipment must be connected to the transformer so that in the event of a fault that would normally trip a breaker/GCFI, an alarm will sound to alert staff to the problem.
A common misapprehension is that an isolation transformer protects equipment against voltage surges, swells, etc. Generally, an isolation transformer provides only limited protection. Over voltage conditions, may well be transmitted through the transformer, and low-voltage conditions almost certainly will be. Spikes (e.g. from lightning) may also be transferred through, although they are likely to be somewhat reduced in energy.
The reason that such confusion exists is because there exist "voltage regulating transformers" (also called ferroresonant transformers). These are specially designed transformers, that are designed to provide a constant output voltage, even when input voltage suffers major surges or dips. They are also good at blocking spikes (e.g. from lightning), and cleaning up distorted power due to nearby heavy industrial machinery. They even provide a very short term (about 0.2-0.3 seconds) UPS like function, which permits them to keep supplying power in the event of a short power glitch (e.g. like power being switched from one line to another). These special transformers are frequently called "isolation" transformers, but that is not their main function.