This is scientifically overly simplified but gets to the point quickly with respect to exploding an atomic device.

Atoms are like little bags of neutrons each trying to break out of the bag and all held together by elastic bands. Put the right number of this little guys in the same bag (say the 143 of them in Uranium-235) and they get real jumpy.

Now you take a neutron that is moving really fast, like a bullet form a very powerful rifle, or expelled from a very powerful explosion and you shoot it at one of those jumpy bags of neutrons all tied together with elastic bands and BAM! a couple of the bands break freeing a few neutrons and leaving behind two smaller bags of say one of Barium and one Krypton (and in this case 3 neutrons are freed).

This splitting of this atom releases some of that energy from the broken elastic bands and fires off the 3 neutrons as if from a powerful rifle or explosion. One of the 3 may fly away, but if there is enough Uranium-235 in the area, it becomes a target for the other two neutrons.

So what started out as one neutron hitting a bag of uranium ends up as a small explosion and 3 neutrons gunning for other uranium... and as long as there is enough uranium near-by this becomes becomes a run-away explosion with each hit firing 3 more neutrons... so 1 becomes 3 becomes 9 becomes 27 becomes 81 becomes 243... until there just is not enough fuel left to sustain the chain reaction.

This is basically it, the physics is a bit more complicated but the short story is one neutron is used to split a large atom into two medium atoms releasing small amount of energy and few more neutrons... and as long as there is enough of the large atoms around the chain reaction will continue in a run-away like flash!