I’ve been wondering about this, but how does the organ of a fire or electric type able to produce said fire or elctricity?

Magic.

On a serious note, for pokémon that have them (because not all fire- or electric-types have specialized organs that generate their elements), it’s really all about physics and chemistry. Fire organs contain chemicals that react to oxygen rather explosively, so the pokémon essentially functions as a literal flamethrower, spraying and spreading fluids that combust in the air and douse a target with fire. Stronger attacks simply involve more of this fluid or a combination of that and other flammable chemicals in the body.

Electrical organs, like a pikachu’s cheek sacs, meanwhile, are modified nerves or muscle tissue composed of thousands of cells known as electrocytes. Think of these like countless miniature batteries, stacked on top of each other. Or if it’s even easier, think of them like regular nerve cells. The difference is that unlike nerve cells (which operate by transmitting minute electrical pulses from one cell to the next), these electrocytes generate an electrical field outward and in larger doses. If you’ve ever noticed your pikachu or other electric-type seem absolutely exhausted at the end of a battle, that’s essentially why: because it actually does take quite a lot of energy (and, no pun intended, electrolytes) to generate the electricity needed for a full-on attack. On the other hand, it’s also why a number of pokémon that have electrical organs also have the ability Static: because it’s not something they can simply turn off.

As a fun fact, while a pokémon’s fire organ has no real analog in the animal kingdom (although many, many animals can spray acid that induces burns comparable to being set on fire), electrocytes aren’t that difficult to find, particularly in fish. The electric eel, for example. It’s just that pokémon have much higher energy reserves and resilience to boot, so they can generate electricity that’s generally more powerful than what you might find among animals. I still wouldn’t recommend picking up an electric eel, though. It’s about as pleasant as picking up a pichu (not that I would know).