I know that the projection function $\pi_{i}(x)=a_{i}$
where $x=(a_{1}, a_{2}, \cdots a_{n})$ is a point in the n dimensional space defined by the cartesian product $A_{1} \times A_{2} \times \cdots A_{n}$ and $a_{i}$ is the $i^{th}$ coordinate of $x$. Source: https://solitaryroad.com/c787.html
Then what does $\pi_{[-c,c]^{n}}(z)$ mean?
where $[-c, c]^{n} = [-c,c] \times\cdots [-c,c]$ and c is always positive.
It's the projection of the point $z$ onto the cube $[-c,c]\times[-c,c]\times\cdots\times[-c.c]$. So, if $z=(z_1, z_2, \ldots, z_n)$ are the co-ordinates of $z$, for each co-ordinate you get $z_i$ if $z_i\in [-c,c]$ and $0$ otherwise. As an example, suppose $n=3$ and $z=(5,-2,1.5)$. Then $\pi_{[-2,2]^3}(z) = (0,-2, 1.5)$
Note that normally a projection is looking at an object with more dimensions that the space being projected to, but it's not necessary