= Solution
Identify $\mathbb R^3$ with the traceless <Hermitian matrices> $X=x\cdot\sigma$, where $\sigma_i$ are the <Pauli matrices>. Then $\det X=-|x|^2$. For $U\in SU(2)$ the action $X\mapsto UXU^\dagger$ preserves trace, Hermiticity and determinant, so gives an orthogonal transformation of $\mathbb R^3$. Since $SU(2)$ is connected, the determinant of this transformation is $+1$.
An element in the kernel commutes with every $\sigma_i$, hence is scalar. Unitarity and determinant one restrict it to $\pm I$. Every three-dimensional rotation is a rotation through some angle $\theta$ around a unit axis $n$. It is obtained by
$$
U=\exp(-i\theta\,n\cdot\sigma/2)=\cos(\theta/2)I-i\sin(\theta/2)n\cdot\sigma.
$$
Indeed multiplication with $\sigma_i\sigma_j=\delta_{ij}I+i\epsilon_{ijk}\sigma_k$ gives the <Rodrigues rotation formula> for $UXU^\dagger$. This proves surjectivity, not just equality of Lie-algebra dimensions. Therefore
$$
\boxed{SU(2)/\{\pm I\}\cong SO(3).}
$$
A full $2\pi$ rotation lifts from $I$ to $-I$, explaining the double cover and the <Spin group> interpretation.
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