Rituximab is a genetically engineered chimeric murine/human monoclonal antibody directed against the CD20 antigen found on the surface of normal and malignant B lymphocytes.  A single, exact mechanism of the anti-cancer effects of rituximab has not been identified.  The current thinking is that rituximab has several anti-cancer effects, most of which involve other components of the immune system.  The effectiveness rituximab may be mediated by the way it works in concert with other therapies, (e.g. chemotherapy), or in conjunction with other components of the immune system, although there is some evidence that it could be effective as a single treatment in some cell lines.  
Rituximab may induce death (apoptosis) of malignant B cells by changes in cell signaling.  Changes that have been identified in response to rituximab in vitro include inhibition of p38 mitogen-activated protein kinase, NF-κB, extracellular signal-regulated kinase 1/2 (ERK 1/2) and AKT antiapoptotic survival pathways.  Another possible mechanism involves complement dependent cytotoxicity, which can result in rapid cell death of rituximab-coated target cells. Also, antibody-dependent cell-mediated cytotoxicity (ADCC) whereby the activity of a variety of effector cells including NK cells, granulocytes and macrophages are recruited by the presence of rituximab on the surface of cells.
Ref:
Weiner GJ. Rituximab: mechanism of action. Semin Hematol. 2010 Apr;47(2):115-23. doi: 10.1053/j.seminhematol.2010.01.011. PMID: 20350658; PMCID: PMC2848172. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2848172/

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2848172/
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