What Is a Mechanical Keyboard?
Basically, different types of keyboards use different technologies to register keypresses. Mechanical keyboards register keypresses through the use of mechanical switches. That's what makes them mechanical keyboards.
Do they do something other keyboards can't do?
Functionally? Not really, no.
Then why bother?
Everybody's answer to this might differ a little bit! But most answers tend to fall into one of the following reasons:
Customizability & Aesthetics the funnest reason for a mechanical keyboard
Mechanical keyboards are endlessly customizable, both inside and out. If you want a keyboard made almost entirely out of wood, it's possible. If you want a keyboard that matches your vaporwave desk theme, it's possible. Here's a rundown on some of the most basic things you can customize:

Layouts, Shapes, & Sizes
Whether you want to try a more efficient layout than QWERTY, a keyboard that is ergonomic and wavy, or a keyboard you can fit in your purse, there's a layout, shape, & size for you!

Keycaps
Keycaps are the switch coverings that you touch when you type! Mix-and-match different colors, try blank keycaps without a legend, or even try keycaps made out of metal on your next keyboard!

Switches
Beneath each keycap is a switch. Some make clicky noises, some give you tactile feedback as you type, and some are silent and smooth. There are dozens of switches to choose from!
The Typing Experience the most satisfying reason for a mechanical keyboard
Most keyboards today are membrane-based. They use a bubbled rubber sheet between the keys and the circuit board instead of mechanical switches. Typing on these keyboards feels mushy and spongey. Comparatively, typing on a mechanical keyboard feels crisp, smooth, and satisfying. It's easier to experience than to explain, sort of like explaining the difference between a '92 beater car and a new Lamborghini. Same function, very different experience.
Nowadays, many of us use keyboards for multiple hours a day. If you're using a keyboard for so long, why not go for a premium experience instead of settling for "functionable"?
Whether you want switches that register each keypress with barely a tap or specific audible feedback with each keystroke, there's a keyboard for your ideal experience.

Durability the hardiest reason for a mechanical keyboard
Since computer keyboards with mechanical switches became popular in the 1980s, they have been known for their reliability. Some of the first mass-produced mechanical keyboards from the 1980s are still beloved, collected, and functionable today. When buying a quality mechanical keyboard, you can be sure it will last with very minimal maintenance.

Community & Enthusisasm the friendliest reason for a mechanical keyboard
Collecting, assembling, customizing, and discussing mechanical keyboards is a hobby that is consistently growing in popularity. Many mechanical keyboard users enjoy the community aspect of owning a mechanical keyboard.

From Twitch streams to subreddits to designer keycap discord chats, there's likely a community you'd enjoy browsing and maybe even participating in!
If community isn't your thing, mechanical keyboards are still fun talking-points to discuss when your nosey coworker asks why you switched keyboards.
Niche Needs the most specific reason for a mechanical keyboard
The ultra-customizability of mechanical keyboards allows them to uniquely address very specific needs. There are switches specifically made for ultra-speedy touch typing. Some keyboards are ergonomic and come in two halves. Some keycaps include all of the shortcuts for programs like Photoshop included in their legends. Whatever your niche, it's likely a mechanical keyboard can better suit it than a normal keyboard.

Special thanks to u/EMCoupling for some inspiration in writing this, and Bernin Uben, Bin Jaleel Almanza, Dustin F Owen, Nubelson Fernandes, National Cancer Institute, Jamie Bradburn, and Girl with Red Hat for the photos used.