When you buy a dedicated server, a someone has to provide your hosting company with bandwidth.
usually hosting companies want to provide their customers with reliable, fast connections. To do this, the get bandwidth from several different places. This is multi-homed. Multi-homed is good.
Single homed is when your hosting company only has one bandwidth provider.
BGP4 redundant means that even if part of the network goes down, people will still be able to see your site. It is like having many different highways going to the same place.