Interesting question! Green tea increases metabolic rate of body because it contains antioxidants known as catechins1. Catechins are actually polyphenol flavonoids, and have been shown by Wang et al and team to affect body composition and decrease body fat in intra-abdominal area. Also, as per Khalesi et al, catechins improve blood pressure, especially when systolic pressure exceeds 130 mmHg, while some studies, such as Ellinger et al, have indicated that catechins may have effect on endothelium-dependent vasodilation which might lead to normal blood flow regulation. Another study, by Weinreb et al, shows that catechins can also reduce the risk of Parkinson's and Alzheimer's. However, their exact mode of action is unknown because of their extensive metabolism during digestion6.
Green tea also contains a little amount of caffeine. Now, as we all know, caffeine is known for its stimulating effects on the body. Caffeine, as stated by Nehlig et al, blocks the formation of adenosine in brain, thus increasing concentration of dopamine and norepinephrine in the brain which, in turn, allow quicker firing of neurons. Along with caffeine, another compound with similar effects is L-theanine, which increases the activity of GABA and leads to anti-anxiety effects, as reported by Nathan et al. Caffeine and L-theanine can together have synergistic effects on the brain and give much more pronounced effects together than individually.
A yet another, and probably less spoken of, effect of green tea, as concluded by Lee et al, is that it can even kill some bacteria and viruses (obviously, killing means preventing growth here) and thus reduces chances of bacterial and viral infections.
Obviously, there might be even more benefits of green tea than I've put here. To the precise point you were asking, we don't yet know how green tea helps in burning calories and fat oxidation. The proposed mechanisms of action are10:
catechins act on catechol-o-methyltransferase (COMT), an enzyme which degrades norepinephrine. Norepinephrine is known for regulating thermogenesis and fat oxidation. Thus inhibition of COMT by catechins may lead to prolonged effects of norepinephrine on fat metabolism.
caffeine can also affect thermogenesis, but by inhibiting phosphodiesterases & thus prolonging effects of cAMP in the cell. As a result, cAMP stimulates hormone - sensitive lipase and promote lipolysis.
catechins and caffeine can also work together and increase thermogenesis in body by 35-43% by increasing fat oxidation.
However, these are just the proposed mechanisms and more detailed studies are required to reach a conclusion.
1. Beneficial Effects of Green Tea—A Review; Carmen Cabrera, Reyes Artacho, and Rafael Giménez; Journal Of The American College Of Nutrition Vol. 25 , Iss. 2,2006
2. Wang, H., Wen, Y., Du, Y., Yan, X., Guo, H., Rycroft, J. A., Boon, N., Kovacs, E. M.R. and Mela, D. J. (2010), Effects of Catechin Enriched Green Tea on Body Composition. Obesity, 18: 773–779. doi:10.1038/oby.2009.256
3. Khalesi, S., Sun, J., Buys, N. et al. Eur J Nutr (2014) Green tea catechins and blood pressure: a systematic review and meta-analysis of randomised controlled trials, 53: 1299. doi:10.1007/s00394-014-0720-1
4. Epicatechin ingested via cocoa products reduces blood pressure in humans: a nonlinear regression model with a Bayesian approach; Sabine Ellinger, Andreas Reusch, Peter Stehle, and Hans-Peter Helfrich
5. Neurological mechanisms of green tea polyphenols in Alzheimer's and Parkinson's diseases; Orly Weinreb, Silvia Mandel, Tamar Amit, Moussa B.H. Youdim
6. "Flavonoids". Linus Pauling Institute Micronutrient Information Center, Oregon State University
7. Caffeine and the central nervous system: mechanisms of action, biochemical, metabolic and psychostimulant effects.; Nehlig A, Daval JL, Debry G.
8. Nathan PJ, Lu K, Gray M, Oliver C.. The neuropharmacology of L-theanine (N-ethyl-L-glutamine) a possible neuroprotective and cognitive enhancing agent. J Herb Pharmacother (2006) 6(2):21–30.10.1080/J157v06n02_02
9. Lee H.J., Lee Y.N., Youn H.N., Lee D.H., Kwak J.H., Seong B.L., Lee J.B., Park S.Y., Choi I.S., Song C.S. Anti-influenza virus activity of green tea by-products in vitro and efficacy against influenza virus infection in chickens. Poult. Sci. 2012;91:66–73. doi: 10.3382/ps.2011-01645.
10. Jówko E. Green Tea Catechins and Sport Performance. In: Lamprecht M, editor. Antioxidants in Sport Nutrition. Boca Raton (FL): CRC Press/Taylor & Francis; 2015. Chapter 8. Available from: https://www.ncbi.nlm.nih.gov/books/NBK299060/