This is the most straightforward version of the canonical proof (i.e. what user26857 posted) that I can think of. I have run into problems, more than once, explaining this proof to skeptics, so hopefully this adds some clarity, rather than more confusion.
Let $R=k[x,y,z]$.
If $R_{\geq 2}$ is the ideal of all polynomials with lowest degree $\geq 2$, and $R_2$ is the $k$ vector space of all homogeneous polynomials with degree exactly $2$, then we can define a $k$-linear map $\pi: R_{\geq 2}\to R_2$ by throwing out the terms of degree $\geq 3$, e.g. $\pi(x^2 + xz + y^3) = x^2 + xz$. It is not hard to see that this map is $k$-linear—it is merely a standard projection.
There is only one property of this projection that we really need: If $f\in R_{\geq 2}$, and $g\in R$ has constant term $g_0$, then $\pi(gf) = g_0 \pi(f)$.
(Proof: $(g-g_0)f$ has only terms of degree $\geq 3$, so $\pi((g-g_0)f) = 0$. Therefore $\pi(cf + (g-g_0)f) = \pi(g_0f) + \pi((g-g_0)f) = g_0\pi(f)$.)
It follows immediately that if $I\subset R_{\geq 2}$ is an ideal, and $I$ is generated by $f_1,\cdots,f_n$, then $\pi(I)$ is generated by $\pi(f_1),\cdots,\pi(f_n)$ as a $k$-vector space. Explicitly: $\pi(Rf_1 + \cdots + Rf_n) = k f_1 + \cdots + k f_n$.
In the particular case $I=(xy,xz,yz)$, we can easily see that $\pi(I)$, the homogeneous degree $2$ part of $I$, has dimension $3$ over $k$. By the above, this means that $I$ cannot possibly be generated by $2$ elements, as they would map, under $\pi$, to a generating set that is too small.
In other words, this property that the minimal number of generators of an ideal $I$, generated by homogeneous polynomials of the same degree, equals $\dim_{R/\mathfrak{m}} I/\mathfrak{m} I = \dim_k I/\mathfrak{m} I$, where $\mathfrak{m} = (x,y,z)$, is an immediate and trivial statement once you are comfortable with this operation of throwing out higher-order terms. And the above generalizes easily.
