Why are support vector machines good at classifying images?
Support vector machines have one built-in "layer" that helps with having an interpretation of the data - the kernel. You could even use output from some other image classifier, including a neural network, as the kernel. E.g. you could measure how far apart two images are in "classifier space" from a trained neural network (maybe one trained against different targets from the ones you want to classify with the SVM)

There is also an implied "layer" in image feature construction.  Common image features are histograms of pixel values, histograms of pixel differences (edge detection), bags of "visual words" and there are many more possible transformations and statistics. 

Typically if you are using a classifier other than a deep neural network (which is supposed to discover the features automatically), then you will pre-process the image into a set of features. That pre-processing covers the majority of an approach being "good at images".