We can assume that immersed surface divides the space into finite number of regions.
Then your formula gives volume of of these regions counted with muliplicity.

[![enter image description here][1]][1]

Subdivide your surface into singular surfaces such that each cuts only one region.
Summing up the isoperimetric inequalities for *some* of these surfaces implies the inequality that you are looking for.


  [1]: https://i.sstatic.net/ENeNi.png