Everytime you introduce a new equation (or constraint) you remove a degree of freedom. For example, consider the 1D complex "vector" $z=a+bi$. It has two degrees of freedom. If I require normalization I introduce the constraint $|z|^2=1$. So I could write my vector as $$z=\frac{a+bi}{\sqrt{a^2+b^2}}.$$ From this equation it is not clear that it depends on one degree of freedom. We can introduce polar coordinates to make it more clear: $a+bi=r e^{i\theta}$. This gives us $$z=\frac{r e^{i\theta}}{r}=e^{i\theta}.$$

To summarize, introducing an equation removes a degree of freedom. This may not be obvious immediately but generally you can reparemetrize to make it clear that a degree of freedom has disappeared.