This is CSG -- Constructive Solid Geometry (although in this case, it's destructive, strictly speaking), if you want to do more research on how the algorithms actually work.
The standard approach is to use the two existing meshes A & B (the house and the subtracted "explosion" volume) to generate a third "mesh" (graph) where the two meshes planes and edges intersect, C -- which describes the area they share. In 2D:
/ A \
| |C| |
--- ----- | ------- G
A is the house, B is the is the explosion volume, C is the union / intersective area between A and B. G is the ground.
The mathematics / logic behind it is not trivial, as there are many edge cases to deal with and half or more of this task is just recognising which of the possible types of outcome graphs are special cases which need to be catered for in code. I've seen the problem described by the veterans on gamedev.net as "a very hard problem" and I can tell you from attempting it in 2D, even, and having a moderate degree of success where only a couple of edge cases fail, that it is no small task.
A way that might be easier is to use a polygon decomposition algorithm, and use the geometry that creates as a basis for your geometry destruction instead.