Sure, there is certainly a lot of jargon here which I could have explained better, but in the end one has to be familiar with some calculus and complex numbers to make sense of this topic. Depending on where you're from, this could be either late high-school to second/third year of university.

So by exponential I mean the exponential function, in the mathematician's sense: something which takes numbers as inputs and produces other numbers as outputs in a unique way. Already a complication: these can be complex numbers.

Then, the derivative of a function is its "instantaneous rate of change". How much does the function increase for small (infinitesimal, really) changes in input? Since this rate may be different for every point, we are defining a new function f': for every input of f, f' gives you the local rate of change. So there you go, the derivative takes a function f and produces a function f'. We call this an operator. Because the derivative has very nice properties, we call this a "linear" operator.

Finally, the "eigen"-things. These are really fancy words for the simple concept of a fixed-point: when you apply an operation to a certain thing X and just get that X back. So, the function x^2 has two fixed-points: 0 and 1. These would be "eigenpoints", but we don't call them that because we reserve the term "eigen" for linear operations, such as the derivative operator or matrix-multiplication.

The point of my answer is that the exponential function is elegantly understood as the eigenfunction of the derivative. In other words: which function is its own derivative? Which one tells you how it changes just by looking at it?