Let $I$ be a homogeneous ideal of $k[X_1,\cdots,X_n]$. The quotient ring $k[X_1,\cdots,X_n]/I$ is then a graded ring: $k[X_1,\cdots,X_n]/I=\oplus_{i\geq 0}k[X_1,\cdots,X_n]_i/I_i$ where $k[X_1,\cdots,X_n]_i$ is the set of homogeneous polynomials of total degree $i$ and $I_i=k[X_1,\cdots,X_n]_i\cap I$. The Hilbert function is the map $HF_I:i\in \mathbb{N}\longrightarrow \mathbb{N}$ defined by $i\longrightarrow dim_k k[X_1,\cdots,X_n]_i/I_i$ where $dim_k$ denotes the dimension of a $k$-vector space.
Is there a not sophisticated proof of the existence of a polynomial $HI_I\in\mathbb{Z}[X]$ such that $HF_I(i)=HI_I(i)$ for $i$ sufficiently large ? ($HI_I$ is called the Hilbert's polynomial). References are wellcome.
In a large sense it depends on what you consider sophisticated. These notes (see Theorem 4) give a nice sketch of a standard proof which you may find easier to follow. This proof is essentially the same as the one contained in "Cohen-Macaulay Rings" by Bruns and Herzog, a standard reference for this theory.