When you talk about the safety function of "breakers", one generally expects to be talking about overcurrent protection. There is also ground-fault (residual current) protection in GFCI/RCD+overcurrent combo breakers, and there is arc-fault protection in AFCI+overcurrent combo breakers.
There are several kinds of safety at work here.
Is generally about life safety. The general idea is that of the two conductors, hot and neutral, one of them is fairly near earth potential because neutral is pegged to earth in the main service. You are unlikely to be shocked if you get between neutral and earth, as there'll only be a volt or two of difference there.
Therefore on a polarized machine, the "hot" part of the machine is made innermost - the least likely for you to come in contact with.
This will not make overcurrent protection more likely to trip. If a faulty polarized machine came in contact with a well-grounded feature like a water pipe, the part coming into contact is more likely to be at neutral potential, which will not make overcurrent protection trip.
It may make GFCI protection trip, and the arcing could cause AFCI to trip.
This is about both life safety and fire prevention.
The idea of grounding is that if the machine has an internal problem, and "hot" goes somewhere it should not be going, the first place it will go is the grounded surfaces/chassis/etc. This impacts life safety because the "hot" goes to ground instead of via a human to another ground. Electricity takes all paths in proportion to their conductivity (1/resistance), and the ground provides a very, very good path which should carry virtually all of the current.
So yes, ground gives a very good path for a faulting device to trip the overcurrent device. You were told correctly.
It impacts fire protection because if it flows enough current to start a fire, the current will be efficiently carried back to the main panel and through the neutral-ground bonding back to source (neutral), allowing high currents to flow, which the breaker will detect and overcurrent trip. In other words, the goal is to turn a ground fault into a bolted ground fault and assure a trip. Without this grounding, a much lesser amount of current could flow, not assuring a trip.
A GFCI breaker will trip on even a very small leakage current. Again, ground helps even a small current find its way back, to assure that the GFCI will trip. By giving current a path to arc against, this also helps an AFCI trip.