Every electron acts like a tiny magnet due to a property called spin(A science word, not actually spinning), and also because it moves around the atom’s nucleus. In most materials, all those little magnetic moments point in random directions, so they cancel each other out.

In metals like iron, cobalt, and nickel, the situation’s different. The electrons interact through what’s called the exchange interaction, which basically makes it more stable for neighbouring spins to line up the same way. When enough of these spins align, they form regions called magnetic domains. Each domain is like a tiny magnet.

In an unmagnetized piece of metal, those domains point every which way, so the fields cancel. When you magnetise it, the domains start lining up, and their combined fields create one strong magnetic field that extends outside the metal.

Sorry, but "theres a lot of tiny magnets inside" is pretty much it. Every charged particle in motion creates a magnetic field. Moving electric charge = magnetic field. So if something contains moving electrons, and every atom does, then at the microscopic level, there’s always some magnetic field being created.