Why convolution over volume sums up across channels?
There are multiple points that I try to explain them. 

First, each filter for convolutional networks for images is a 3d volume. Consequently, whenever it is said we have $n$ filters, means we have $n$ volumes of those 3d filters. 

Second, you can consider each convolutional layer as an MLP which is applied to small regions of the input. These are applied to different regions of the input to investigate whether a typical pattern is in that region or not. These patterns are going to be learned by means of cost functions. You can easily consider that for each filter which is a volume, you are concretely doing a summation over weighted inputs, exactly as MLPs. 

Third, as the result, your trained filters will decide to choose the information of which channels depending on the task using the cost function. They may be in a single plane or they may be among multiple of them.