We don't 'need' to know any of these words. But if you're talking in Welsh, you may end up mentioning one of these places, just as you may end up mentioning jam, or Wednesday, or football.
The "need" to know the name Caergrawnt is a lesser one, perhaps (although it certainly does come up in sports reports), but Amwythig is really quite important: you'll hear it in train departure announcements on Welsh railway stations (yes, it will be announced in English too, but the Welsh announcements precede those in English!) and it also appears on road signs in Wales (above "Shrewsbury"). I'd say it's just as important as -- say -- knowing that the Austrian city of Wien (Vienna) is called Bécs in Hungarian.
Ah, but I'm not so sure that the "they" at Wikipedia are all that well informed, allintolearning! :)
"Fortified", yes, but in a very particular way. The town of Shrewsbury has an English name (ending in Old English burh) that does indeed denote a fortified place (characterized -- see the first part of the name -- by a prevalence of scrub/shrubs), but the Welsh is more particular.
The historic centre of the town of Amwythig / Shrewsbury is almost completely encircled by a large bend in the River Severn, making it a near island, so that the (g)wyth (= ditch / channel / firth) of its Welsh name describes its moat-surrounded (amwythig) situation very precisely!