What is meant by "continuity" is if we place the probes of our meter (this is a piece of test equipment such as this (
http://www.fluke.com/Fluke/usen/Digital-Multimeters/Fluke-77-IV.htm?PID=56126), one on each end, then we should read 0 ohms or a very small number if the meter is capable, i.e. 0.001 ohms if the fuse element (the little wire or ribbon of metal inside the glass envelope) is good. If a fuse is "blown" or has opened due to overload then the element will have burned or melted and is no longer going to pass voltage and current through it. Our meter will read "OL" or open loop. The fuse will no longer pass voltage and current through it and the circuit is effectively dead.
I give you all this in detail because as it was stated earlier you can't always just look at a fuse and see that has blown. Some fuses may blow inside the caps and you just don't know for sure so you test it to make absolutely sure. This has bitten me on the butt before and not just troubleshooting my audio gear but at work too. I always test a fuse unless it clearly has burn marks inside the glass, usually accompanied by spatters of the melted metal in with the burn marks.
This all sounds kinda like overkill but when a line is down at my plant it is costing us anywhere from ~$8K to $20K an hour for downtime and it is all due to a 5 cent fuse. There have been instances where the electrician on duty missed a blown fuse for an entire shift and when asked if he tested the fuse for continuity the answer was "No, it looked fine to me" you can be sure that person got a couple of extra days off without pay so he could think about it.
Hope this helps but if it adds confusion then let me know and I'll try to explain it better.