Intuition behind Hilbert's Nullstellensatz

Solution 1:

May be you will find this method unnatural as well, but go to The Stacks Project, browse to the chapter "Exercises" (this is chapter $74$), and take a look at exercise $10.1$. You can also look at Spectrum of a linear operator on a vector space of countable dim in which I ask a question related to $10.1$. Note that this proves the Nullstellensatz only for $\mathbb{C}$, but has the advantage of using the language of linear algebra which you may prefer more/ find more intuitive.

The other option would be to convince yourself that Noether Normalization is saying something geometric (the Nullstellensatz is an easy consequence of Noether Normalization), and for this I can recommend Ravi Vakil's "Foundations of Algebraic Geometry" (found here) sections $11.2.3$ to $11.2.6$ in the March 23rd version of the notes, although this may be an overkill.

Ofcourse, @mbrown's comment of rephrasing the Nullstellensatz is perhaps the best way to think about it. Good luck!