Once upon a time (before the 20’s), belts were mostly for decoration or military use; the average gentlemen would hold his trousers up with braces. As men started to wear their trousers lower, the braces were gradually exchanged for belts; but braces have a several advantages over belts which have ensured that they are far from obsolete.
To start with, they are more comfortable; because the braces hang the trousers from your shoulders, they (the trousers) are typically a little wider than trousers cut for a belt, meaning that, not only do they always hand in the right place, they are far more comfortable when sitting. Also, there is no uncomfortable belt, clamped around your stomach.
Not only that, but you never have to worry about your pants falling down. Even with a well-fitting belt, the constant motion of walking, sitting and standing can cause your trousers to slip down; this is never an issue with braces.
Finally (and for some, most importantly), braces make your look better. Let me explain; first, particularly with trousers that don’t fit perfectly, when you wear a belt, you are pinching fabric around your waist, which looks bad. Braces eliminate this problem, and allow your trousers to hang perfectly straight in the process. Secondly, if you have a bit of extra girth in the stomach, there is no belt to draw attention to that area, which should help make the weight less noticeable (in theory). Finally, because you are not wearing a belt, there will be less of a distinction between your legs and your torso (depending on the contrast between your shirt and trousers), which can help you appear taller. This is why in general braces are highly recommended for short, stocky men.
Now, on to actually wearing the braces.
The most common type of attachment mechanism is the clip, but the proper one to use is either the leather button tab, or the fabric button tab (for black and white tie).
The clip type is the easiest to use, simple clip them on to your trousers. The button tabs require your trousers having buttons sewn into your waistband; something that you are unlikely to find in off-the-rack trousers, but are easy enough to add yourself. Some companies will even provide buttons with their braces, like The Tie Bar.
While button on braces are preferable, the only absolute rule is that you don’t wear braces with a belt, like a rube (I say rube now).
If you are ready to make the switch from a belt to braces, it’s a simple procedure to remove the belt loops from your trousers and add buttons; this is the very first thing that I do to a new pair of trousers. If you don’t want to do it, then take them to a tailor and have them do it; better yet, have the tailor make you a pair of trousers.