Mudguards (typically called fenders in the US) are a great feature and come in many varieties. The fenders you have are full-coverage fenders and are likely aluminum. Metal fenders are still quite common (and in my opinion, fenders are an essential addition to any bike if you live where it rains. Velo Orange manufactures many sizes and styles that are affordable.)
The fenders are held to the bike via fender stays (long aluminum rods) and are mounted directly at the fork crown in the front and the brake bridge and chainstay bridge/bottom bracket shell in the rear.
Fender stays are usually mounted to threaded bosses (with an M5 or M6 cap screw) integrated into the front and rear dropouts (or near the dropouts.) In situations where you don't have dedicated bosses, you can use p-clamps to mount them securely.
At the other end of the stay there is a nut and an eye bolt which allow the distance between your tire and the fender to be adjusted (10mm minimum!) Note: There are many different methods for attaching fender stays, but this is what your fenders utilize.
To install, you simply put the eye bolt onto the fender stay, push the threaded part through the hole on the mounting tab, and tighten the nut down to secure the stay. Do everything loosely at first, then adjust the distance from the fender to the tire and center it.
Here is a link to an online vendor, but most bike shops will have this hardware on hand.