I understand that for each of N particles they have a position:
$$\vec{r}_i=\vec{r}_i (q_1, q_2, \dots, q_n, t)$$
where $q_1, q_2, \dots, q_n$ are n generalized coordinates. However I don't understand why each generalized coordinate $q_i$ is a function of positions. ie.
$$q_i=q_i (\vec{r}_1, \vec{r}_2, \dots, \vec{r}_N, t)$$
I need examples on how it is possible for a generalized coordinate to depend on positions.
Your second equation is the inverse relation of your first equation.
Given a set of values for $(q_1, \cdots, q_n)$, you can compute the values of $(r_1,\cdots, r_n)$ according to the first equation. If you know the values of $(q_1 \cdots, q_n)$, you can compute the values of $(r_1, \cdots, r_n)$ according to the second equation.
The Cartesian coordinates and generalized coordinates are two equivalent sets of coordinates. You can describe the system using only either one of the sets. You can also switch the representations.
You may want to do calculations mainly in the generalized coordinates. However, you may be given the initial data in the Cartesian coordinates and required to output the results in the Cartesian coordinates. Then, you need to go back and forth between the two representations by these coordinate systems. The first equation lets you do so in one way, and the second equation is for the backward transformation.
For an example, it may suffice to consider 1 particle on a plane. The Cartesian coordinates of this particle may be written as $(r_1, r_2) = (x,y)$, and the generalized coordinates may be written as $(q_1, q_2) = (\rho, \theta)$. As the example of the first equation, we may define \begin{align} x =& \rho \cos\theta, \\ y =& \rho \sin\theta. \end{align} This set of generalized coordinates are called the polar coordinates. If we are given a initial position of the particle in the $(x,y)$ representation, then we want to know the corresponding values of $(\rho,\phi)$ in order to do some calculation in the generalized coordinates. For this purpose, we need to solve the above relation for $(\rho,\phi)$. We obtain \begin{align} \rho =& \sqrt{x^2 + y^2}, \\ \theta=& \arccos\left(\frac{x}{\sqrt{x^2+y^2}}\right). \end{align} [You may need to take care of the fact that multiple values of $\theta$ gives the same value of $\cos\theta$ (and $\sin\theta$), i.e., multivaluedness of $\arccos\theta$ (and $\arcsin\theta$) if you really want to implement the calculation.] You may then solve the equation of motion in $(\rho,\theta)$. If your potential does not depend on $\theta$, the equation of motion will be simpler in the $(\rho,\theta)$ representation than in $(x,y)$ representation. Finally, you may be required to report the position of the particle at a given time in the Cartesian coordinates. Then, you can calculate them through the first equation [i.e., $x=x(\rho,\theta)$ and $y=y(\rho,\theta)$] in this answer.
[Your equations include time ($t$) explicitly. For an example of this type: consider a boat running along a river. If you construct a set of generalized coordinates such that $(q_1,\dots,q_n,t)=(0,\dots,0,t)$ corresponds to the center of this boat at all $t$, then the two equations between $r$ and $q$ depend on time.]
Edit
Regarding the comment about the vector notation under the original question: a vector can be a function of scalers, and a scaler can be a function of a vector.
The first equation may be written as \begin{equation} \vec{r}_1(\rho,\phi) = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \rho\cos\phi \\ \rho\sin\phi \end{pmatrix} . \end{equation}
The second equation may be written as \begin{align} \rho(\vec{r}_1) =& \sqrt{[\vec{e}_x \cdot \vec{r}_1]^2 + [\vec{e}_y \cdot \vec{r}_1]^2}, \\ \theta(\vec{r}_1) =& \arccos\left(\frac{\vec{e}_x \cdot \vec{r}_1}{\sqrt{[\vec{e}_x \cdot \vec{r}_1]^2 + [\vec{e}_y \cdot \vec{r}_1]^2}}\right). \end{align}
where \begin{align} \vec{e}_x =& \begin{pmatrix} 1 \\ 0 \end{pmatrix},\\ \vec{e}_y =& \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \end{align} Note that \begin{align} \vec{e}_x \cdot \vec{r}_1 =& x, \\ \vec{e}_y \cdot \vec{r}_1 =& y. \end{align}
