I need help clarifying what (R1,..RN) is in this context Let R be the space of all the N! permutations of (1,2, ..., N), and let R = (R1,...RN) be uniformly distributed over R.
The thing i don't understand is what "R = (R1,...RN) be uniformly distributed over R." mean

Your statement will make a bit more sense if you use more standard notation to differentiate vectors of values from sets.  I would write it as follows:

Let $\mathscr{R}$ be the space of all $N!$ permutation vectors of $(1,2,...,N)$ and let $\mathbf{R} = (R_1, ..., R_N)$ be uniformly distributed over $\mathscr{R}$.

This means that the vector $\mathbf{R}$ is a permutation vector of the set $(1,2,...,N)$ and this vector has $N!$ possible values that are collected together in the set $\mathscr{R}$.  To give a simple example, for $N=3$ you would have $3!=6$ possible values for the permutation vector, given explicitly by the set:
$$\mathscr{R} = \{ (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1) \}.$$
The uniformity specification then means that:
$$\mathbb{P}(\mathbf{R} = \mathbf{r}) = \frac{1}{N!}
\quad \quad \quad 
\text{for all } \mathbf{r} \in \mathscr{R}.$$
