Rotations in three-dimensional space can be represented by the usual real 3 x 3 matrices. They work on real 3 x 1 column matrices which represent a vector. By doing so they yield the representation of the rotated vector. But rotations can also be represented by complex 2 x 2 matrices working on complex 2 x 1 column matrices. This is the representation SU(2). The complex 2 x 1 column matrices do here no longer represent vectors but rotations. They contain the information about the Euler angles, but also (by equivalence) the information about the rotation axis and the rotation angle, or the information about the image of the triad of basis vectors under the rotation. In fact, one can consider them as a steno for the 2 x 2 matrices by writing only their first column because the second column is unambiguously defined by the first column, and this property is preserved under multiplication with SU(2) matrices. In fact the matrices of SU(2) are of the form:
$ a \quad -b^{*}$
$b \quad \quad a^{*}$
The spinor is just the first column of that matrix.
The 2 x 1 spinor matrices are normalized to 1, in conformity with the definition of SU(2): $aa^{*} + bb^{*} =1$. They therefore contain the equivalent of three independent real parameters, i.e. the three Euler angles, or the unit vector along the rotation axis plus the the rotation angle, etc...
The question remains how we put the information about the rotation into these 2 x 2 matrices. That is done by writing the rotations as a product of reflections. (It is easy to verify that the product of two reflections is a rotation. The intersection of the reflection planes is the axis of the rotation. The angle of the rotation is twice the angle between the reflection planes. The reflections are thus generating the group of rotations, reflections and reversals).
Such a reflection matrix ${\mathbf{A}}$ is easy to find. One uses the unit vector ${\mathbf{a}}$ perpendicular to the reflection plane. The coordinates $a_x, a_y, a_z$ of ${\mathbf{a}}$ must occur somehow in the reflection matrix but we do not know how. Therefore we write the matrix heuristically as $a_x {\mathbf{M}}_x + a_y {\mathbf{M}}_y + a_z {\mathbf{M}}_z$. The matrix ${\mathbf{M}}_x$ will tell where and which coefficient $a_x$ will occur in ${\mathbf{A}}$. Analogous statements apply for the matrices ${\mathbf{M}}_y$ and ${\mathbf{M}}_z$. E.g. if the matrix ${\mathbf{M}}_z$ is
$ 1 \quad \quad 0$
$0 \quad -1$
then the matrix ${\mathbf{A}}$ will contain $a_z$ in position (1,1) and $-a_z$ in position (2,2). The same is true, mutatis mutandis, for the matrices ${\mathbf{M}}_x$ and ${\mathbf{M}}_y$. To find the expressions for these three matrices we express that a reflection is its own inverse, i.e. ${\mathbf{A}}^{2}=1 $ where $1$ stands for the unit matrix. This condition can be satisfied if the matrices ${\mathbf{M}}_j$ satisfy the conditions ${\mathbf{M}}_{x} {\mathbf{M}}_{y} + {\mathbf{M}}_{y} {\mathbf{M}}_{x} = {\mathbf{0}}$ (cycl.) and ${\mathbf{M}}_{x}^{2} = 1$, ${\mathbf{M}}_{y}^{2} =1$, ${\mathbf{M}}_{z}^{2} =1$. In other words, the matrices ${\mathbf{M}}_j$ can be just taken to be the Pauli matrices $\sigma_{x}, \sigma_{y}, \sigma_{z}$. The matrix becomes then:
$ \quad a_z \quad\quad\quad a_{x} -\imath a_{y}$
$a_{x} + \imath a_{y} \quad \quad -a_{z}$
Once we have the reflection matrices, we can obtain the rotation matrices by multiplication. This leads to the Rodrigues formula. The spinors are just the first columns of these rotation matrices. If you want it in more detail (and exactly along the lines explained here), you can read it in the third chapter of "From Spinors to Quantum Mechanics" by G. Coddens (Imperial College Press). You will there also find the link with isotropic vectors mentioned by KonKan, and with the stereographic projection.
This shows that a spinor is a way to write a group element. That idea remains valid when you want to develop the group representation theory for spinors of the homogeneous Lorentz group in Minkowski space-time (with a few complications, among others due to the metric; Instead of three Pauli matrices you will now need four 4 x 4 gamma matrices. If you know the Dirac theory, you will recognize that this procedure is exactly the way Dirac obtained the gamma matrices. But he used the energy-momentum four-vector $(E,c{\mathbf{p}})$ to define the basis rather than the four unit vectors ${\mathbf{e}}_{\mu}$ of space-time. It is all explained in detail in the reference above).
As there is a 1-1-correspondence between a set of basis vectors and the rotation that has produced it by operating on the canonical basis, you can thus visualize the spinor as a rotated basis. This remains true in Minkowski space time (now with a set of four basis vectors and with Lorentz transformations). In ${\mathbb{R}}^{3}$ we have ${\mathbf{e}}_{z} =
{\mathbf{e}}_{x} \wedge {\mathbf{e}}_{y}$, such that ${\mathbf{e}}_{x}$ and ${\mathbf{e}}_{y}$ are actually sufficient to reproduce the whole information. You can therefore also represent the complete information by the isotropic vector ${\mathbf{e}}_{x} +\imath {\mathbf{e}}_{y}$. After rotating ${\mathbf{e}}_{x} +\imath {\mathbf{e}}_{y}$ you can just find
${\mathbf{e}}'_{x}$ and ${\mathbf{e}}'_{y}$ by taking the real and imaginary parts of the rotated isotropic vector ${\mathbf{e}}'_{x} +\imath {\mathbf{e}}'_{y}$, and thus reconstruct the full rotated basis. This is the reason why one can also represent the rotations by using isotropic vectors. The rotated isotropic vectors are in 1-1-correspondence with the rotations.
It is not any more difficult than that. It is simple geometry. There should not remain any secret or mystery that stands in your way to fully understand this. It is never well explained, which is probably due to the extremely concise way Cartan introduced them, without explaining what is going on behind the scenes. I hope this helps. Kind regards.
