How could HIV spread to so many people, if the contraction rates are?
It's really rather simple: HIV has been spreading for a very long time, and people have more sex than you think they do. The chances of transmission per sexual act are low, but they scale rapidly as you have sex more times, and most people, shocking as this may sound, have sex more than once. As with any disease, the infection rate increases exponentially with time. And HIV has been spreading for a very long time. It first jumped to humans in around 1908 but was not recognized for three quarters of a century because, as you say, the transmission rate is very low. It didn't hit the interesting part of the exponential curve until the 1970s. This is how exponential growth works.