The difficulty with the question is the word intuitive. Intuition basically reflects our understanding of the world around us, which is described by classical physics. Quantum mechanics is exactly the regime where our intuition breaks down because it functions very differently from the world of our everyday experience. As Terry Pratchett said :
It’s very hard to talk quantum using a language originally designed to
tell other monkeys where the ripe fruit is.
It is exactly that difference that we're using to get the computational speed-up.
There is a sequence of standard algorithms that most quantum computing texts progress through: Deutsch's algorithm, Deutsch-Jozsa, Simon's/Bernstein-Vazirani. These are chosen because they are the easiest to understand. They all have broadly the same structure, but increasing complexity, with a corresponding gain in computational speed (with Simon's giving exponential speed-up). You will not understand them intuitively. You have to do the maths. I think the closest that you will come is through the following explanation of Deutsch's algorithm:
Imagine a one-bit function $f(x)$. Either $f(0)=f(1)$, or it does not. Your task is to determine which. Obviously, in the classical world, you have to evaluate $f(0)$ and $f(1)$; two function calls. In the quantum world, crudely speaking, you can look at both values simultaneously, and perform a one-qubit measurement (which will give you one bit of information), but you can choose that measurement so that the one bit is a global property of the function, in this case $f(0)\oplus f(1)$. The same is broadly true of the other algorithms I mentioned: there is information, due to the structure of the problem, hidden in the collective properties of the function evaluations, and it is those collective properties that you are trying to determine.