In Indian cooking, what does it mean when a recipe says "until the oil begins to leave the side of the pan"?
Since I'm not an expert in Indian cuisine, I can't say this for certain, but I agree with the comments so far that this seems basically equivalent to the separation of the oil from the curry.

To my mind, the wording makes perfect sense, because in some cases with a lot of oil/butter, you can get separation happening early on the pan's edges.  Oil tends to float on top, and usually the center of a pan is heated more strongly over a burner.  Thus, you tend to get boiling/bubbling in the center, which tends to push any floating oil toward the edges.

But these recipes are not looking for that small amount of oil along the edge (which, depending on stirring and the thickness of the curry, can stay there through much of the cooking process).  They are specifically waiting for the time when separation increases and fat appears floating on the surface even away from the edges of the pan.