"Just a few days ago I had our grease trap pumped clean and today the trap overflowed. Why did this happen?"
Hopefully you had it pumped before it was stopped up. Pumping the grease trap alone does not prevent stoppages down the line. While pumping grease, grease can get into your main line if the pumping is done wrong. Also, if you had it pumped because you had a stoppage and so assumed it was due to the grease trap, it might be that you had a partial stoppage already in your main line.

"What connections come at the both ends of your grease trap interceptors?"
Both ends come with no-hub fittings. In other words, they come with a plain end. To adapt the ends to steel or plastic pipe you simply use a standard no-hub adapter.

"If I buy a new one from you will it backup and clog less than my current grease trap?"
Grease traps back up either because they need to be cleaned or because of an outlet blockage. Buying a large unit would mean you'll need to clean it less often as there is more storage capacity of grease. But please understand that all grease traps will clog if you don't clean the grease regularly.

"How do grease traps work?"
They work by slowing down the flow of warm/hot greasy water and allowing it to cool. As the water cools, the grease and oil separate and float to the top of the grease trap. The cooler water (now with much less grease) continues to flow down the pipe to the sewer. The grease itself is actually trapped by baffles, which cover the inlet and outlet of the tank, preventing grease from flowing out of the trap.

"Will these grease traps prevent future blockages?"
Grease traps do help prevent stoppages after the trap but blockages can (and do) occur as a result of insufficient or infrequent grease trap line maintenance. The grease must be removed from the trap; a grease trap or drain line which is not periodically maintained will eventually clog.