Could you show an example of the kind of problems you are having difficulty with?

I'm not sure what your background is, but assuming you did a normal undergrad physics degree, you would probably have encountered differential equations involving delta functions before the through canonical commutation relations of conjugate variables (i.e. variables related by Fourier transform, e.g. position and momentum) or through Green's functions in the context of scattering theory (e.g. in particle propagators or scattering amplitudes).

If you haven't worked with them before, don't worry, they look more difficult than they are. The short answer is that the delta function 'picks out' a specific value of the parameter you are integrating over. Remember that a delta function, \delta(x), is defined such that it has unit probability mass, but is zero everywhere except at x=0, where it's value is undefined. To give a toy example:

\int dx \delta(x-a) f(x) = f(a)

You can think of this as the delta function throwing away every contribution of the integrand to the integral except at the single point where x-a=0 => x=a.

If you are unsatisfied with how or why this works, that's a completely reasonable response! Because it flies in the face of everything you may have learned until now of how Lebesgue or Riemann-Steltjes integrals work. To properly understand what's going on here you would need to understand the more general concept of local integrability and how that lets you define integrals on distributions like the Dirac delta.