Firstly in short, we need to know that an electron is a particle which is negatively charged and revolves around nucleus which is positively charged. Both of them together forms the building blocks of matter which means the atoms.
Secondly, we require some knowledge about uncertainty principle. According to uncertainty principle any microscopic particle (which includes particles like photon which makes light, electron, positron etc) does not have distinct velocity and precise position (obviously in space) simultaneously. Higher the accuracy in measurement of velocity, lower would be the accuracy in measurement in its position and vice a verse.
Now if electron falls on the nucleus its position and velocity will become fixed simultaneously (meaning we can easily measure that it is on nucleus and its velocity is zero). This contradicts the uncertainty principle (as stated above) and hence electron does not fall into the nucleus of atom.
If you think this is too easy and lame then here goes another. If electron fall on nucleus, the measurement in its position will become highly accurate. In other words, the error in measurement is will be less. If this is so, then uncertainty principle will cause large error in its velocity. When its velocity will increase (it cannot decrease as electron cannot fall further into the nucleus) the electron will gain large kinetic energy which will cause it to escape electrostatic force.
This is the simplest reason I found out. It is from the book ‘Six Easy Pieces’ by Richard Feynman (a scientist who I think is greater than Einstein himself)