In more common terms, people generally think of staking as a type of riveting. It is a way to make a joint between two pieces of metal. In staking one of the pieces has a post (called a boss) that passes through a hole in the other piece. The boss is slightly undersized, so that will slip through the hole. Once the boss is in place, it is smashed out in such a way as to completely fill the hole and press tightly against the hole all along its length. The hole usually also has a countersink, and the boss is smashed out so as to fill the countersunk hole. When making a trigger guard, the boss is usually just smashed out with a hammer. (If working on small delicate item, like a watch or silver cup, a special tool called a staking punch may be used. The special punches help the smith avoid damaging the surrounding surface.)
Strictly speaking, a staked joint is different from a simple rivet. A rivet is not part of either piece, but is instead just an independent piece of rod that links the two pieces together. And in making a simple rivet head, the boss is not necessarily made to fit tightly along its length.
In making a staked lug, the smith will saw and file the lug so as to make a post (boss). He will drill a hole in the extension of the trigger guard. The boss should be able to slip through with a gentle tap from the hammer. The guard is then flipped upside down and the lug is securely clamped in a vise. The boss is then smashed out with a series of blows. Once the boss is smashed out so as to completely fill the countersink, then the excess can be filed off smooth with the surface of the guard.
The lugs on both extensions of the guard are made the same way.
I’ve always (only) done this with the pieces cold. This seems to help keep the hole from expanding as the boss is smashed. I can imagine situations in which it might be done hot, but I’ve never needed to do it hot when working with something as small as a trigger guard. Maybe other’s have different experience. (Comments anyone?)
Some people worry about the boss slipping or turning in the hole, and so they think they need to braze or silver solder it in place, but that’s not really necessary with a trigger guard. The lug is wedged up inside the stock and both ends of the guard are pinned down, so the guard and lug never experience any torque. The joint might break if you had to use the rifle as a club to fight off a grizzly (hmmm)… but in that case you would have more serious problems anyway! Provided it has a good countersink, it usually never budges.
Here’s a sketch. I’ve drawn the guard right-side-up, but keep in mind that most of the work is actually done with the guard upside down. (It’s just easier to understand if you can view it right-side up.)