You can motivate the methods somewhat - if that weren't the case, no one could have thought of them. I can't usefully explain without an example, so I apologize if the math that follows bores anyone.
One of the standard methods is "integrating factors for first-order linear equations". You are told that, faced with an equation
y' + p(x) y = q(x) you should multiply both sides by e^(the integral of p(x)).
For example, you might have
y' + (2/x) y = x.
Then you multiply by e^(integral of 2/x), which is x^2.
[Sometimes I wish Hacker News had TeX available.] If that's all you tell people, it looks like some random abracadabra and it's no wonder why people feel they just don't get it. So you might try to explain this way:
"The equation has a derivative in it. To undo a derivative, you need to integrate. But if you integrate as-is, you have no idea how to integrate y' + (2/x) y."
"Well, you know that the integral of df (the derivative of f) would be just f. So if you could make the left side look like the derivative of something, then you could just integrate both sides."
At this point, you scratch your head and think: "What could I do to make the left side be the derivative of something?" This kind of thought is impressionistic - you have to think in a vague way of things the left side "is like". Daydreaming for a while, you might realize it's a sum of two terms, so you think: If this is a derivative and it's the sum of two terms, what derivative rule gives a sum of two terms? And you might think of the Product Rule.
But the given thing is not the derivative of a product as is. What to do? So continuing this line of thought, you might think - maybe I can multiply it by something to make it the derivative of a product. Once again, you have to search through your experience with derivatives and maybe mess around on scratch paper. Finally, you realize x^2 works - multiplying by x^2 makes the equation
x^2 y' + 2 x y = x^3.
The left side is d(x^2 y), so you can integrate both sides and get x^2 y = (1/4) x^4 + c.
The final step is to think whether you can generalize what you did with "p(x)" instead of "2/x". After some additional messing around, you come up with the integrating factor I gave at the start.
I have no idea who discovered this method, or what their thought process was (if they even explained it at the time). This was about the extent of the motivation I got when I was taught this stuff in high school/college. I'd tell students this sort of thing when I taught differential equations. But I don't know what other people do in teachng, and I'm not sure this helps. For people who feel their differential equations courses were baffling/unmotivated, is this the kind of explanation you want? Or do you want something completely different, like applications?
At some point, explanation ends. Can a painter say why he put a daub of paint of that color in that place in a painting, or can a writer say why he had a character do this or say that? There are several points in the motivation above where all I can say is "you have to sit there and think and mess around", even after paragraphs of writing. I'm not sure how to do better.