In this article what does I represent in this equation?
As an aside for clarification, does the block matrix at the end of the equation mean "a new matrix with I spread out in the lower right, -1 in the top left, and 0 filling the remaining spaces on the top row and left column"?
In matrix notation, particularly with no other context, $I$ almost always represents the identity matrix of the appropriate size, i.e. a diagonal matrix whose entries are all equal to 1. Likewise, if $0$ is used in reference to a matrix it is the zero matrix of corresponding size. So in this context, where $\mathbf{u}$ is an $n$-vector, and so $\mathbf{uu}^T$ is an $n \times n$ matrix, in the expression
$R = I - 2 \mathbf{uu}^T \begin{pmatrix} -1 & 0 \\ 0 & I \end{pmatrix}$
the first $I$ is the $n \times n$ identity matrix, and inside the block matrix the second $I$ is the $(n - 1) \times (n - 1)$ identity, and the $0$s are suitably sized vectors of $0$s. To put it another way, the block matrix is actually just the $n \times n$ identity matrix with the first entry multiplied by $-1$.