(Altitude x 3) OR (Distance x 3) comes from the fact that the ideal descent path of an aircraft is 3 degrees. Further, most ILS approaches, if not all, have a glide path angle of 2.5 to 3.25 degrees.
If you use 3 degrees as an average, what you will find using trigonometry is that for a given distance (NM), multiplying that distance by 3 will give you the height in FT. As an example, let's use a 3 degree profile and a ground distance of 10NM.
Tan(3) = Height/10NM
Tan(3) x 10NM = 0.524NM = 3183ft
As you can see, we need to make approximations and have to round numbers up or down - but even with the approximation and simple maths in your head, you will see that in real life it works quite well.
Using trigonometry you can see that the (Altitude to lose x 3) to give you distance required works quite well also.
Now this distance/altitude relationship is directly to touchdown. For instance, if you are 10NM away, you should be at 3000ft and descending at the appropriate descent rate to achieve the 3 degrees from your present position to touchdown. But as some of the others have already mentioned, you need to take into account your airspeed as well as the headwind/tailwind component. For instance, it is no good being at 3000ft at 300KTS and with a 20KT tailwind. Technically, you are still on the "correct" glide path as calculated but given the situation, it is unlike that you will achieve a desirable outcome.
This x3 method doesn't just work for distance or altitude to touchdown. It also works for meeting altitude restrictions on your way down. For example, (height to lose x 3) will give you the distance from the waypoint in which you need to begin your descent. Equally, checking if you are high or low to meet a altitude restriction can be done by using (Dist to waypoint x 3).
You will find that these are general rules of thumb, and are not meant to be exact. But they work surprisingly well and using these methods will bring you pretty much on profile when you join the glide slope.
Hope this answers your questions somewhat!