Well, the reason is that everybody's strings need changing at different times. Some reasons are because (1) there are so many kinds of strings and they age differently; (2) people all play differently and their strings wear out (or corrode) faster or slower as a result; and (3) in different parts of the country (or the world) strings are just going to need changing more often.
So, what do you look for in deciding whether to change them? One thing can be obvious: buildup of corrosion or gunk on the string. This really happens a lot in warm, humid climates. If the buildup can't be removed with a little steel wool, then it'd definitely time to change strings! (When I was starting out, this happened on my mandolin strings every few days.)
Another sign is when the strings get hard to tune. Often it's because they're not sliding smoothly through the string-nut (that's the little white thing with slots at the bottom of the peghead). If you put on new strings, and when you're at it, put a little graphite -- pencil-lead dust will do-- in the bottoms of the little nut-slots, then the tuning should get a lot better.
Another sign of elderly strings might be that they don't play in tune. If you're pretty sure that your bridge is in the right place, but your banjo is still "noting out" more than usual up the neck, then new strings might be what you need.
One more sign of old strings may not be as obvious. If the instrument (banjo or otherwise) just doesn't sound right, the strings may have gotten too old to sound good at all. When does this happen? Well, this is the most extreme case of old strings, since it may take several months or a year for the strings to get this old.
Some players take extreme steps to keep new strings on their instruments, especially if they break a lot of strings. Back when we were playing a lot of festivals, I used to change the strings on both mandolins and both guitars every morning before we played our first set. That was a lot of work, but it helped keep the string-breakage to a minimum. Others take a different approach. I've heard that Bill Monroe changed his mandolin strings once a year, at New Year's, and from then on just changed them as they broke (which they did, pretty often).
Now, this all applies to the fretted instruments. Fiddle strings seem to fall into a different category. I've known fiddle players who changed their strings every few months, but as for myself, if the fiddle gets new strings every five years, that's a lot. I suspect that the strings on my fiddle now have been on it for longer than that!
So the answer to the question is, that it's up to you yourself to decide when to change strings. There are a lot of reasons for changing them (better tone, volume, and tuning), and there are plenty of reasons for just leaving them on there (less hassle with awkward work, and less risk of getting your banjo or mandolin bridge out of place in the string-changing process, among other things). But if you go in for a lesson and your teacher takes one look at your strings and turns as green as they are, then it's time.