The same way one gets better at fencing: practice, practice, practice after lessons, lessons, lessons.
First off, if you are not familiar with the Fencing Officials web site already then you should be. Go through the study materials there to make sure you know what you need to know above and beyond describing the conversation of the fencers. You are in charge of equipment, safety, behavior, and many other things aside from determining a touch. The official documentation will also teach you the correct hand signals and verbiage to describe a phrase.
Check with your local clubs and division to see if they run referee training or camps. As almost every division needs directors for tournaments, there is usually some effort at the division level to train people.
Volunteer to direct bouts at your local clubs. If possible (and ask for permission), video the bouts you direct and compare your calls against slow-motion replays later on.
Go to all the divisional level tournaments you can and offer to help out either by directing (if you are confident you can) or by assisting a director by keeping time or charting scores. But your real purpose is to watch and learn (granted, if there are no rated directors at the tournament, this doesn't work as well).
Go to all the regional or national tournaments you can but pay more attention to the directors than the fencers. If possible, talk to the directors before and after the tournament and ask them questions about a call. As they usually are not the focus of attention, you might find a lot of them very receptive and willing to answer.