Behrend's theorem regarding AP states that
There exists an absolute constant $c$ such that for all sufficiently large integers $N$ there exists a subset $A$ of $\{1, 2, \cdots, N \}$ with at least $Ne^{-c \sqrt{\ln N}}$ members such that no three elements of $A$ forms an arithmetic progression.
The proof doesn't uses any heavy machinery except typical high school math. The two key steps of the proof is:
Step - 1: Let $\displaystyle S_{n,r,m} := \{ (x_1, \cdots, x_n) | 1 \leq x_i \leq m \; and \; \sum_{1 \leq i \leq n} x_i^2 = r^2 \}$. Then you can pick a suitable integer $r$ such that $|S_{n,r,m}| > \frac{m^{n-2}}{n}$.
Step-2: You can define an injective function $f: S_{n,m,r} \rightarrow \{1, 2, \cdots, (2m)^n \}$ such that if $f(x), f(y), f(z)$ forms an AP, then the vectors $x = y = z$.
It's not hard to fill up the details once you know of the following two above steps (use PHP for Step-1, and explicitly define $f(x_1, \cdots, x_n) = \sum_{i=1}^{n} (2m)^{i-1}x_i$ for Step-2)
But I find considering the steps to be highly unmotivated. In particular, I have some questions:
[1] When the bound $Ne^{-c \sqrt{\ln N}}$ is replaced by $Ne^{-c \ln n} = N^{1-\epsilon}$ (for fixed $\epsilon$), you can give an explicit construction by picking $A = \{a_1, \cdots, a_k \}$, where $a_i = \lfloor (i+C)^{1+2\epsilon} \rfloor$ for some big constant $C$. But for the original problem, what's the motivation behind considering the existential arguement instead of an explicit construction ?
[2] Suppose I agree with you and believe that an existential arguement would do the proof. Then one way I can do this is to consider the existence of a function $f(x): \mathbb{R} \rightarrow \mathbb{R}$ for which $f(\mathbb{Z}) \subset \mathbb{Z}$ and a line intersects the graph of $f(x)$ at atmax two points, and then let $A = \{f(C), f(C+1), \cdots, f(C+k) \}$ for some constants $C, K$. But from where the idea of considering lattice points on the $n$ dimensional sphere come from ?
[3] Even after I agree with you on the motivation behind [1], [2], how you come up with those weird inqualities on size of $S_{n,r,m}$ and size of image of $f$ which magically finishes the proof and gives the bound of $Ne^{-c \sqrt{\ln n}}$ at the end ?