This question pops up every now and then. Really, why? Aren't people just happy with what they've got? Isn't just installing and deinstalling fonts the usual way good enough? Here are several reasons.
Designers working by contracts are doomed to use a variety of fonts, because every client is special (well, at least that's what clients like to hear, don't they?) and has his/her own preferences and ideas about what suits and what doesn't suit a particular task; because design tasks are actually different. They really are.
So we are talking here about collections of fonts that are as large as 3.000 fonts. Or 5.000 fonts. Decorative freeware fonts, licensed fonts, very expensive licensed font families with over 20 faces and so on. Keeping all of them loaded in the system is likely to slow it down and make browsing through fonts lists a living hell. Which means you need some tool to activate and deactivate fonts selectively.
OK, so convinient activation and deactivation. What else? Searching. If you have over 1K of fonts you might as well want being able to find the font you need as soon as possible given that usually you have only a basic idea about what you need. So you want to compare them: type in some text and see how all candidates render this text, then pick the one that suits the job best.
So, that's it? No. How about grouping some fonts? Think of branding projects. You create a whole package — website, brochures, business cards etc. — and use a fixed set of fonts that you worked out after hours of being shouted at by a highly paid manager in an expensive suite. At any time you have to be able to quickly enable these fonts in the system to create a new brochure or a new bunch of business cards for this client (big companies do reorder them once a month or two). Font managers are here to help you by providing tagging system.
Now imagine that you work for a font foundry... Okay, okay, you hate these big companies, they just don't understand the digital age of typography... yadda-yadda... Whatever. So, your job is to promote fonts that you do. The most convinient way is to create a so called font book — an album displaying how same text is rendered using some fonts. You want it now and you want it quick. Font managers are your friends again: you filter the whole collection to select just the fonts you want in the font book and in few more clicks you have a PDF file ready to be uploaded or printed.
And the last bit. If you ever used complex design software, you know that sometimes it finds a corrupt font and refuses to work. The traditional way to fix this is to remove one half of the collection and try again. If the application is still refusing to load, you take away half of this half and try again. So after spending an hour of your precious time on this stupid research you find the blasted .ttf, bang it against trash bin and think: "How could I possibly avoid waisting my time on this the next time?" Unbelievable, but the answer is font managers again: they often have built-in font validation tools that help finding problematic fonts and deactivating them temporarily.
All in all, if you are graphic or font designer, you just need a font manager. And if you are just a user who got this fontmatrix package in his Linux distribution installed by default and are wondering now what on Earth you are supposed to do with it, you probably have a good idea by now.