Intuition behind the ILATE rule

The way I see it, when you differentiate an inverse trigonometric function, you don't get another inverse trigonometric function. Instead you get "simpler" functions like $1/(1 + x^2)$ or $1/\sqrt{1-x^2}$. This does not typically happen with the antiderivative of such functions.

Similarly, when you differentiate a logarithmic function, the logarithm disappears.

So, when using integration by parts $\int u dv = uv - \int v du$, it makes sense to select the inverse trigonometric or logarithmic function to be the one that is the $u$ term.

In the case of algebraic, trigonometric and exponential functions both integration and differentiation don't change the nature of the function, so they come later in the ILATE order.

Of course, this is just intuition and there are examples where you can violate this so called rule and still integrate by parts without any problems.

As a technique for explicitly integrating functions given by formulas as usually seen in calculus classes, integration by parts works because $u'v$ can be easier to integrate than $uv'$. There are multiple ways an integrand can be considered as a product of the form $u'v$, but some choices lead to $uv'$ being of no use, perhaps even harder to integrate. The ILATE mnemonic you mention (or whatever it is) gives some rules that approximate the best guesses as to what will work well. Better than this mnemonic (which I've never thought about) is to just have enough experience doing integrals to be able to see what will work and why (at least for the type of integrals one typically sees in a calculus class).