Example 5, Given a cube $A B C D$ $A_{1} B_{1} C_{1} D_{1}$ with edge length 1 inscribed in sphere $O$. Find the surface area of sphere $O$.
Slightly explained: Draw the section through the diagonal plane of the cube (Figure 7), where the diagonal $A C_{1}=\sqrt{3}$ is the diameter of the sphere. Therefore, the surface area of the sphere $S=4 \pi\left(\frac{\sqrt{3}}{2}\right)^{2}$ $=3 \pi$.