Nuclear bombs were invented due to a series of advancements in physics that occurred in the late 1800s and early 1900s that revolutionized our understanding of the world. I will treat this as a list of the most important ones

1. Electromagnetism. In the 1800s, a great deal of research was done on electricity and magnetism that ended up realizing that these two forces are the same force in different forms. The main result of this for our purposes is that this work led to the discovery that the speed of light is constant and dependent on the laws of physics, which implies that if the laws of physics stay the same regardless of where you are, the speed of light is constant. This led to a major problem in physics since this is kind of impossible in classical physics.

2. Atomic theory. In the 1890s and 1900s the structure of the atom began to be understood. Scientists discovered that atoms are made of a nucleus comprised of positively charged protons and uncharged neutrons, and negatively charged electrons surrounding it.

3. Radioactivity. Around the same time as modern atomic theory was getting going, scientists such as Marie Curie began studying radioactivity and the properties of radioactive substances and the fact that radioactive elements can turn into other elements

4. Relativity. Albert Einstein revolutionized physics by solving the problem that arose in step 1 where light has a constant speed regardless of where you are, by developing his Theory of Relativity. Essentially Einstein used math to determine what the laws of physics would be like if we just accepted that the speed of light in a vacuum was just constant regardless of any other effects. This resulted in drastically different physical laws than what we may expect based on Isaac Newton's formulation of physics, but importantly the differences are only apparent in circumstances we don't see very often in our everyday lives. Most important to our nuclear bomb discussion is that a weird result of Relativity is that mass and energy are equivalent to each other, via the famous equation E=mc^2. This means that matter can be converted into energy and vice versa. This also explained why radioactive substances can turn into different elements only by emitting energy.

5. Nuclear science. In the following decades, scientists used the previous discoveries to develop a broader theory of nuclear science, gaining a better and better understanding of how radiation works, how to control it, and how to produce it.

After all of that, we finally had the theoretical background for a nuclear bomb. Nuclear bombs work by converting a relatively large (i.e. more than a couple micrograms) amount of matter into energy very rapidly. In practice, this means putting a lot of material that can trigger a chain reaction of radioactive decay events all in one place. Nuclear bombs are designed very carefully to use explosives to very rapidly push together nuclear material, which creates the chain reaction that leads to nuclear explosion. The reason the amount of energy is so great is that same E=mc^2 equation. E is energy, m is mass, and c is the speed of light. Since the speed of light is a very big number, you can see that even a fairly small amount of mass is equivalent to a quite large amount of energy. If you can convert a significant amount of mass into energy very quickly, it leads to a giant explosion because of the sheer amount of energy released.