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Without the 1:1 isolation transformer, if you touch the chassis (which could be at 120 volts on your TV) and also touch any other grounded point, you will have 120 volts AC across a portion of your body. Other grounded points include the case of your oscilliscope, the ground of your multimeter (if not battery operated), the case, if metal, of the lighting you may be using, etc. Anything with a ground/neutral connection to the power line.
Even if you do get a shock yourself, connecting a scope ground to the chassis which is at 120 volts will cause a direct short across the 120 volt line, possibly damaging the scope and your shop wiring.
With the 1:1 isolation transformer, a hazard would occur only if you touched both sides of the isolated supply. As one side of this is used to generate your TVs DC power supply , it is usually contained in the TV and an accident is much less likely.
With the 1:1 isolation transformer, touching terminals and surfaces of your test equipment, lighting, etc will not cause an electric shock or damage test equipment.
When working on un-isolated equipment, it is very important to have this transformer. In fact, I believe that it is much more important than the variac.
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