Those charts only show pressure - higher pressure prevents evaporation and convection so we get little or no rainfall. Lower air pressure means that evaporation and convection is much easier, allowing more rainfall.
Take this for example from the Met Office. This shows (roughly anyway) where the boundaries of cooler and warmer air masses are. That blue line with triangles over the UK is a cold front which brings cloud and rain, followed by cooler conditions. If that were a warm front, warmer conditions would follow.
Hope this helps (and please somebody point out if I make a mistake, so I can blame my Geography teacher who calls me Victor Meldrew =])