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They didnt, atleast not directly. Although e=mc^2 is often associated with the production of thermonuclear weapons, the formula was just the equation Einstein derived from theory to prove that thermonuclear weapons were theoretically possible.

Energy = mass x (the speed of light)^2.

It basically says that enormous amounts of energy can be released from a relatively small amount of material, but says nothing about how it is to be released or what material can be used.

E=mc2 is a principle that relates matter and energy.

Basically it means that a small amount of energy can be converted to a massive amount of energy. In fact the energy you can get from the mass you have is = the mass you have times the speed of light squared, or E=mc2

In a nuclear fission reaction, which is what is happening in an atomic bomb, a very small amount of matter is converted to energy.

I'll give you some number to give you an idea. If you were to convert 1 kilogram of matter (a litre of water) into energy you would produce:

E = 1kg x 3x10^8m/s

= 90000000000000000Joules

= 90 000 teraJoules

That is a a lot of energy. Typically only a very small amount of matter is converted, not 1kg, but you can see how astronomical it is!