That is BS. IANARS, but the orbit that supply- or crew-vehicles sent to the ISS are launched on is probably the equivalent of a Geostationary Transfer Orbit, except that you want to end up in the low-earth-orbit of the ISS instead of the geostationary one. This means that they are launched more or less on an elliptic orbit, with the high point of the ellipse intersecting with ISS's circular orbit. At this high point, you do a 'circularization burn', after which you are at the same height and same speed as the ISS. I am for sure skipping over some details, such as orbital inclination, but there is no fundamental reason why you can not launch at exactly the right time so that you are really close to the ISS just after this burn.
One reason why it takes several days might be due to launch inaccuracy: there are always small errors in the orbital parameters just after launch, so you probably want to allow for some time to adjust the orbit with small burns. Another reason might be procedural, you might want to do some potentially dangerous checks of your vehicle before you come close to the ISS. As an example of how short a rendezvous can be, the Russians recently launched and docked a Progress freighter in six hours (instead of the usual 2 days), they plan on doing this in future with the manned Soyuz.