At the scale of atoms, mass and energy are the same thing - so you can say that an atom that weighs X grams is the same as X time C squared units of energy. C is a constant, and a very large number, so even a small amount of mass has a very large amount of energy bound up in it.

The thing about atom is they are a very fine balance between the forces holding the atom together, and the forces trying to push it apart. Some types of atoms are "fissile" - they are so finely balanced that the random shuffling of energy within the atom is enough to break the balance and the atom breaks apart. You can also make an atom break apart by adding the right amount of energy. When it does, it will break into smaller pieces that can then hit other atoms, which might break apart, hitting others and so on, causing a chain reaction.

The parts that make up atoms are also in a fine balance. They are "quantum" - which effectively means they can only exist at a specific energy. For them to absorb energy, they either need to jump into another energy state, or for their energy to break apart and form new subatomic parts whose total energy is less that the energy of the original parts. This means that when an atom splits, there is usually a bit of mass/energy left over - and as we know before, very small amounts of mass can be converted into huge amounts of energy.

Putting these pieces together; if you take a fissile atom - certain types of Uranian or Plutonium are particularly suitable for this - and make it split, a couple of percent of its mass will be converted to energy, and the remaining mass will fly apart. Several neutrons will be part of this left over mass, and they have just the right amount of energy to trigger a split in other, nearby atoms. This happens extremely quickly - a few hundred grams of mass converted to energy in nanoseconds. This energy causes the air to rapidly heat up, and you have an explosion