How to see citation contexts of all the papers that cite a paper?
From my experience, I think ResearchGate is the only platform currently offering exactly that. Wherever there is a paper fully available in its database being cited by another fully available paper, it shows an excerpt of the citation. Example below:

"... The number of cases where these approximations are used is vast. For instance, among many, 1[2][3][4][5][6][7][8] for the single- or few-mode approximation, or, e.g., [9][10][11][12][13] for the usage of 1+1D cavities. While simplifying the problem, some- times the rationale for these simplifications remains to be justified, above all in relativistic regimes. ..." 

Reference: Interaction of a moving two-level atom with a single-mode quantized cavity field

Citing: [6] Relativity and quantum optics: accelerated atoms in optical cavities

The above was copied from the platform's citation display of a random reference. 

I believe this is an interesting and powerful feature, however it would rely on everyone having a ResearchGate profile plus all references uploaded to the platform. And I do not see that happening any time soon.

Hope that helps? Good luck.