Why Laplace-Beltrami operator is so popular for 3D shape analysis.?

Because it provides many useful properties for shape analysis. If you view the 3D shape as "samplings" of a Riemannian manifold, then you get an associated Laplace-Beltrami operator (LBO). Knowing this operator, or an approximation of it, lets you do many interesting things, known as "spectral shape theory" and "diffusion geometry".

For instance, if you compute the eigenvalues and eigenfunctions of the LBO, you can get a "signature" of the shape. This has many powerful properties, such as being isometry invariant and "similarity" (meaning small perturbations in the shape lead to small perturbations in the eigenvalues). (See for instance: Reuter, Martin, Franz-Erich Wolter, and Niklas Peinecke. "Laplace–Beltrami spectra as ‘Shape-DNA’of surfaces and solids." Computer-Aided Design 38.4 (2006): 342-366.) Any of the "good" discrete LBOs approximate the continuous one of the underlying manifold (or converge in the point cloud case).

Another popular signature is the "autodiffusion" heat kernel, which is directly derived from the LBO eigenvalues and eigenfunctions. It can act as a local and multiscale signature, so it can work well even on partial shapes. See: Sun, Jian, Maks Ovsjanikov, and Leonidas Guibas. "A Concise and Provably Informative Multi‐Scale Signature Based on Heat Diffusion." Computer graphics forum. Vol. 28. No. 5. Blackwell Publishing Ltd, 2009.

Thus, spectral descriptors let you do shape retrieval and matching, even for partially missing or articulated shapes.

If you are computationally oriented, spectral shape analysis has been used for fairly sophisticated computer vision and machine learning tasks; for instance:

• Masci, Jonathan, et al. "Geodesic convolutional neural networks on riemannian manifolds." Proceedings of the IEEE International Conference on Computer Vision Workshops. 2015.
• Litman, Roee, and Alexander M. Bronstein. "Learning spectral descriptors for deformable shape correspondence." IEEE transactions on pattern analysis and machine intelligence 36.1 (2014): 171-180.