These statements are not true, simply speaking. Currents as little as few microampers can kill a person if, for example, applied directly next to the heart. A few seconds of 220V can kill a person even if the current is in miliamper range (Biksom, cited below, mentions 4s at 120mA). However, it is true that it is not the voltage alone that kills, that there are other factors (most importantly, the effective current, which, however, normally increases with voltage).
Look, Ohms law states that R = V / I (resistance equals to voltage divided by current). Given a constant resistance, a higher voltage produces also a higher current, full stop. Given a voltage of, say, 24 V, and the resistance of our bodies being 1000 - 4000 Ohm, the current will not exceed 6 - 24 mA. Note that in case of 220V this largest current would be less than ten times these values, so well below 1A! Yet, this is sufficient to kill an adult.
What's more, extremely high voltage can actually break the skin and effectively lower the skin resistance, which results in an even higher current (see formula above).
So where does this popular knowledge come from? Well, we know that things like electric sparks can have very high voltage, but are harmless. They are harmless not because the current is low, but because the duration and the total energy is low, since the total energy is given by power (in watts) times seconds, and power P = V * I (voltage times current).
Also, it comes from anecdotical evidence, since being shocked by thousands of Volts and survive can make you famous... precisely because it is extremely rare to get into contact with a high voltage line which can supply a steady flow of the current and survive.
Another reason became apparent in the comments: confusion between the factual current of the jolt one is getting and the nominal amps on the fuse. The former is given by the resistance and voltage; the higher the voltage, the higher the current. However, an 1A fuse is likely to blow quicker than a 10A fuse, thus cutting the line before the current penetrates our body to reach the heart and bring it into fibrillation.
It is a similar story with lightnings: duration is extremely short, but more importantly, in most of the cases the lightning did not run through the person who survived -- because if it did, they would be dead (also, see this fascinating bit). It either run over the skin, seeking the path of the lowest resistance, or through a material nearby.
Also, high frequency currents are less likely to penetrate the skin.
Biksom, "A review of hazards associated with exposure to low voltages"