1. The excess glucose is turned into fat. This is a causal link with obesity.
2. To turn the excess glucose into fat, insuline spikes (there is also an insulin index, which can be different than the glycemic index). This is a causal link with developing diabetes (the insulin insensitivity variant).
3. The associated glycemic crash (which is very significant) causes a desire to eat more, especially more quickly digestible food. Again, a causal link with obesity.
In a relatively nutrient-poor environment the effects make complete sense: Get as much of the cheap energy providing stuff as possible and store the energy for later. In the Western world it leads to issues.