One intuitive way to understand a DAE is to interpret it as a dynamical system which can be controlled by some input signals, whose output signals have to satisfy some (equational) constraints. For a typical multibody system, the input signals are the forces perpendicular to the constraints, the output signals are the positions of the bodies, and the (equational) constraints on the output signals are fixed distances between the bodies.

The input signals must now control the dynamical system in such a way that the output signals always satisfy the constraints. This is difficult for a multibody system, because the forces only control the rate of change of the velocities, and the velocities only control the rate of change of the positions, while only the positions must satisfy the constraints.

Reducing the index is easy in theory, because if we assume that the positions satisfy the constraints at the current time instance, then we can just replace the constraints on the positions by constraints on the velocities ensuring that the positions will continue to satisfy their constraints. In practice however, we don't want to throw away the constraint on the positions after we determined the constraints on the velocities, but we do have to throw away some of the initial (differential) equations, if we don't want to end with an overdetermined system.

Determining the constraints on the velocities from the constraints on the positions might be tedious in practice, but at least it is straightforward (and canonical) once you understood the principle. The constraint $c(y,t)=0$ implies $\frac{d}{dt}c(y(t),t)=0=\frac{\partial c}{\partial y}*\frac{d}{dt}y+\frac{\partial c}{\partial y}$. This is not an (equational) constraint yet, because $\frac{d}{dt}y$ is not a variable but only the derivative of a variable. But the other differential equations allow us to express $\frac{d}{dt}y$ as a function of the variables, in our case $\frac{d}{dt}y=v$ for $v=\dot{y}$, so we get the equational constraint $0=\frac{\partial c}{\partial y}*v+\frac{\partial c}{\partial y}$ (or rather $0=\frac{\partial c}{\partial y}*\dot{y}+\frac{\partial c}{\partial y}$ if you manage to not get confused by using $\dot{y}$ as a variable instead of the derivative of a variable).

Throwing away some of the initial (differential) equations is less straight forward (or canonical). If we can use a constraint equation like $y_1^2+y_2^2=1$ to determine $y_1$ as a function of the other variables (i.e. $y_1(t)=\sqrt{1-(y_2(t))^2}$ in this case), then we can throw away the differential equation for $y_1$, i.e. a differential equation of the form $\frac{d}{dt}y_1=\dots$. But we might have also decided to throw away the differential equation for $y_2$ instead, because the constraint also allow to determine $y_2$ as a function of the other variables. But no matter how easy it is to throw something away, this can easily destroy some symmetry of the system we didn't want to destroy, or we might be forced to switch which equation we throw away during the numerical simulation and thereby introduce undesired artifacts. So this part makes index reduction really challenging in practice.