We present a combined numerical and experimental study of Andreev scattering from quantum turbulence in superfluid $^3$He-B at ultralow temperatures. We simulate the evolution of moderately dense, three-dimensional, quasiclassical vortex tangles and the Andreev reflection of thermal quasiparticle excitations by these tangles. This numerical simulation enables us to generate the two-dimensional map of local Andreev reflections for excitations incident on one of the faces of a cubic computational domain, and to calculate the total coefficient of Andreev reflection as a function of the vortex line density. Our numerical simulation is then compared with the experimental measurements probing quantum turbulence generated by a vibrating grid. We also address the question of whether the quasiclassical and ultraquantum regimes of quantum turbulence can be distinguished by their respective total Andreev reflectivities. We discuss the screening mechanisms which may strongly affect the total Andreev reflectivity of dense vortex tangles. Finally, we present combined numerical-experimental results for fluctuations of the Andreev reflection from a quasiclassical turbulent tangle and demonstrate that the spectral properties of the Andreev reflection reveal the nature and properties of quantum turbulence.