In those reaction one or more products might be not soluble in org layer, and well soluble in aqueous layer, therefore as the products move to the next layer, according to La-Chatteleior Principle the overall rate of the reaction increases. Thats why Phase transfer catalysts are used in many reactions to improve the yield which are apparently low yielding.
Actually, if you are asking about the interface, then the answer is MUCH more complicated, take a look at: S. Narayan, J. Muldoon, M. G. Finn, V. V. Fokin, H. C. Kolb, K. B. Sharpless, Angew. Chem. Int. Ed. 2005, 44, 3275–3279.
The interface is an environment where the energy of the binary system have an excess due to nature os interactions. The formation of a transition state at that place may improve kinectics of reaction as well as stabilizes energetics of the entire system, by formation of products.
For detailed information please read: Why Does Water Accelerate Organic Reactions under Heterogeneous Condition? J. Phys. Chem. A, 2013, 117 (12), pp 2446–2454.