Why is $P = I_N - \vec{1}\vec{1}^T/N$ a projection matrix, and $P^2=P$? Why is $P = I_N - \vec{1}\vec{1}^T/N$ a projection matrix, and $P^2=P$?
For example, for N=3 :
$$P = I_3 - \vec{1}\vec{1}^T/3 = \begin{pmatrix} 0.67 & 0.33 & 0.33 \\ 0.33 & 0.67 & 0.33 \\ 0.33 & 0.33 & 0.67 \\ \end{pmatrix} $$
and,
$$P^2 = \begin{pmatrix} 0.44 & 0.11 & 0.11 \\ 0.11 & 0.44 & 0.11 \\ 0.11 & 0.11 & 0.44 \\ \end{pmatrix} $$
But the text I am reading says P is supposed to be a projection matrix, for which $P^2 = P$ ?

Your computation of the matrix is wrong. 
$$
P = I_3 - \frac{1}{3}\vec{1}\vec{1}^T = \begin{pmatrix} 0.67 & -0.33 & -0.33 \\ -0.33 & 0.67 & -0.33 \\ -0.33 & -0.33 & 0.67 \\ \end{pmatrix}
$$
Note the negative signs. '
This is indeed a projection; it sends $(1,1,1)$ to $(0,0,0)$ and sends $(-1, 1, 0)$ and $(-1, 0, 1)$ to themselves, so in the basis consisting of those three vectors, it's projection onto the second two coordinates. 
