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?
but in the end one has to be familiar with some calculus and complex numbers to make sense of this topic.
I think this is more important than the listing of terms one by one in some of the sibling comments. The intros to Calculus, Analysis, Linear Algebra you get in high school/university will get you to the right place. But if you're unfamiliar with 'function' or 'exponentiation', you're probably missing some more secondary-school-level maths background to tackle those topics. That's not a difficult gap to close but it's a necessary first step before taking the next one.
>> ... in the end one has to be familiar with some calculus and complex numbers to make sense of this topic.
> I think this is more important than the listing of terms one by one in some of the sibling comments. The intros to Calculus, Analysis, Linear Algebra you get in high school/university will get you to the right place.
That's certainly true, but if someone wants to pursue understanding of these, simply saying "You need more familiarity with ..." gives them nowhere to start, and no terms to search for. Trying to give some context for each item, and to mention some of the underlying words, gives them a place to start, should they choose to spend the time investigating further.
Right, in the next bit I said I just don't think the person asking the question yet has the mathematical tools to tackle either those courses or most of the things you'd get from searching the terms and they'll need to get some more basic stuff out of the way first.
The self-guided search from terms is obviously fine and it's way more than fine to offer to personally give some stranger on the internet pointers as you did.
It's the one solution that satisfies the initial value condition y(0)=1.
This idea is analogously applied to the case of systems of ODEs with constant coefficients. The system y'=y=Iy has a basis of solutions of the form (e^x 0 ... 0)^t, (0 e^x 0 ... 0)^t, etc. Written column-wise into a matrix, this gives us the Wronskian Y which satifies Y(0)=I, and this Y is exactly the matrix exponential e^(xI).
More in general, the equation y'=Ay has the Wronskian Y=e^(xA) which satisfies Y(0)=1.
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?