The “intensity” in High “Intensity” exercise, refers to the degree of effort that comes from carrying a “set” of an exercise to the point where no further repetitions are possible despite maximum effort to do so. There is a threshold of effort that must be reached in order to elicit a “growth response” for your body to adapt and get stronger. While going to complete “failure” is likely not necessary to elicit this response, it is very difficult to measure “close to failure” as many people are not truly aware of what their maximum effort can provide.
It is commonly reasoned, that “the last rep is the safest rep” because at that point, your muscles have been sufficiently fatigued so that even a maximum effort produces very little actual force, and so it should be next to impossible to exceed the structural integrity of connective tissue.
However, if, in a misguided attempt to complete an additional repetition, a trainee, falls into bad posture, muscle spasms and other problems can occur. This is why constant supervision is vital to ensure that “failure” comes at a point where no additional “perfect” repetitions are possible. If we see form compromised, the set is immediately terminated.