Why there are nodes in orbital?
It is simply because the wave nature of the functions.

Imagine a tight rope with fixed ends (like a string in a guitar). If you shake it it will vibrate. Both ends remains fixed. If you take a high frame rate video and see it in slow motion, at a fixed horizontal length (let say $x$ positions) the rope will oscillate up and down with some amplitude. It can be seen one or more intermediate $x$ positions that remains always with null amplitude (if it is in state close to a stationary one). There you have nodes in one dimensional string.

Visual example

You would have more complex patterns going to a two dimensional system, and even more in a three dimensional case. They all have nodes, like the 1D case. The 3D case can be represented mathematically by using the spherical harmonics. The angular part of the solutions for the hydrogen atom are those spherical harmonics. The only intuitive explaining that I found is their wave nature. Of course it is not as visual as in the 1D case which is easily found in the daily life. And clearly, what is more weird is the wave nature of matter.

I hope that this answer address your question.