Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory.
It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line.
We can therefore say that integral geometry in this sense is the application of probability theory (as axiomatized by Kolmogorov) in the context of the Erlangen programme of Klein.
The content of the theory is effectively that of invariant (smooth) measures on (preferably compact) homogeneous spaces of Lie groups; and the evaluation of integrals of the differential forms.
Subsequently Hadwiger-type theorems were established in various settings, notably in hermitian geometry, using advanced tools from valuation theory.