This is a very good question – and one that many of us in autism research are trying to answer! The truth is that right now we don’t really know for sure.
Many people who have autism can speak very well (though some start talking later than those without autism), but about 25% of autistic people cannot speak – or have only a few words. This percentage is much better than it used to be (it was 50% a few decades ago) and this might be due to the fact that we can detect autism earlier and start intervention programs sooner. Nevertheless, there are still a lot of autistic people who don’t manage to develop any language, and we need to figure out why.
She, and many others, believe that it may be the social aspects of language that are particularly problematic for autistic people, and that this may be what makes speaking difficult. In addition, it might be that joint attention skills like pointing and shared eye gaze (where you follow someone’s gaze to look at what they are looking at) are necessary for the development of spoken language. We know that joint attention is reduced (and sometimes absent) in autism and this may be preventing autistic people from learning to speak.
On a neuroscientific level, some say that the brains of verbal and non-verbal autistic people are different from each other. Very recently (a couple of weeks ago), a paper was published that suggests that brain activity in toddlers and infants with autism might show their capacity for language later in life. The study showed that for toddlers with autism that went on to develop good language skills, the neural activity in language-sensitive areas of the brain was similar to that of non-autistic toddlers. On the other hand, in autistic toddlers who later showed poor language skills, there was reduced brain activity in those language-areas. While this finding is very interesting, we should be careful as the results were based on only one brain scan per child. More research is needed to build up a clearer picture of any brain-based differences that exist between those autistic individuals who develop language and those that do not. You can see the paper here: http://www.cell.com/neuron/abstract/S0896-6273(15)00219-6