How weapons work is complicated. We have different ways we can consider "how weapons work."
The Overview of Weapons
Humans devise tools to enhance their abilities. Weapons are just the tools humans devised for harming other animals and people. Some of them are explicitly for that purpose, such as many swords and polearms, while others have utilitarian uses as well, like camp knives or hatchets. The end goal for weapons is the same: do enough damage to a target(s) such that they cannot participate in combat or function any longer.
How Damage is Done
Weapons, especially of the non-explosive and non-chemical variety, generally rely on breaking bones, disabling muscles (by cutting or thrusting), or decreasing blood supply (bleeding) to achieve "damage." Each weapon will have varying capacities for each of these modes of damage-dealing. Understanding what a weapon is good at and relative advantages against an opponent is also very important to being an effective martial artist.
Frequently, weapons achieve these modes of damage by generating a lot of energy and focusing forces. Points and even blades act to focus forces, weight and center of mass manipulate how quickly it can accumulate kinetic energy and how much of that goes into the target (instead of, say, rotating the weapon after impact). Some other considerations exist, too, like recovering from a blow and not getting stuck in the target. In any case, weapons generally allow for more focused forces and/or quicker energy generation. This is one of the main advantages over hands and legs. Consult the Strain-Energy Equation if you want a more rigorous approach.
Of course, interrupting other life-critical systems is also an option (such as beheading, paralyzing, poisoning, etc). These are not as reliable against opponents who can resist. So I am just acknowledging that they exist, not the primary methods.
How a Weapon is Used
There are many, many techniques for using weapons. Enough for a lifetime of study and practice. There is also an equally large number of weapons to train and to master. There are some concepts that run through all of them, like distance, timing, and technique. In this respect, there is no one way that weapons work.
Even something like a European longsword (or "bastard" or "hand-and-a-half" sword) has multiple grips and literal books worth of techniques! Each grip and technique caters to one or several situations. There are good times to use them and bad times, and that's the "art" part of martial art.