This is a rule of thumb that applies very broadly. Most fitness programs target 12 weeks because that is where anything (again, speaking broadly) that is done about every other day generally approaches an asymptote, with a significant fraction of the change occurring in about half that time.If you start weight training say 3 times every week. You will see something quite fast, already after 6 to 8 weeks. Weight training shapes up the body. Giving the illusion that more fat was lost than what actually shows on the scale. Tight and toned muscle takes up less space than loose, flabby muscle mass.
When my dad became a body builder back in the 50s he said that twelve weeks made a huge difference. When I got serious about my aerobic ability when I was 18 it took right at 12 weeks to go from not being able to jog half a mile to doing 3 miles nonstop in about 26 minutes. And that's where things stayed unless I really up'ed the effort. My dad said the same thing -- with roughly the same amount of effort that he started at he could maintain the changes that he achieved in the first twelve weeks, but to make significant progress beyond that required a higher level of effort.
At this stage of my life I am not looking for the kind of high-performance in my fitness that I might have always dreamed of. I know that there are people that start marathon or Ironman type training in their 50s, but I've concluded that I don't have the desire to put in that kind of effort. But I'm also convinced that what I can reasonably achieve in 12 weeks at a modest level of effort is so far beyond where I am now that it is a worthy goal. I would like to get to the point where I can ride a bike on ten-mile long rides, jog three miles, and fly a kite with my daughter (and generally be active with her). On a more ambitious note I would like to get so that I can hike Colorado's 14'ers (at least a few of them) with my family and perhaps take up my karate training (which I really enjoyed and got pretty got at -- when I was 16-19 years old!). I don't care about ever competing again, but if I could gain the kind of flexibility that I had then (with expectations toned down for what is reasonable for a 50 year old), it would be wonderful. If I do get my weight down to where it is feasible to take up karate again, I plan to put in a reasonable amount of effort and accept the results that that level of effort produces.