Disclaimer: None of the ontological positions detailed is my own.
1. Neurosciences and the category mistake: a first take
As Peter Reynaert puts it in Reynaert, P. (2015): "Neuroscientific Dystopia: Does Naturalism Commit a Category Mistake?" In: Neuroscience and Critique (pp. 70-86). Routledge (this is a synopsis of his):
My claim that naturalism commits a category-mistake has consequences for assessing neuroscience. Neuroscience is a variant of naturalism which reduces aspects of consciousness and human existence to brain processes. The identification of this mistake has both ontological and methodological consequences. The absurdity of naturalism implies that human consciousness and existence are conceptualized with notions and theories that cannot be applied to them, because they do not belong to the ontological region called nature. Second, and because of this, the scientific methods and theories developed for nature cannot be used to elucidate and understand human existence. Contemporary debates about the (un)reality of the free will for instance, whereby arguments against its existence are allegedly borrowed from neuroscientific research, are simply futile, because they illustrate par excellence the absurdity of the naturalist approach.
This does not necessarily commit him to substance dualism, but this is a basic argument the position often is built upon. I would paraphrase and summarise it as follows: We cannot just discard our phenomenal and normative reality by applying explanations and concepts that are methodically and conceptually tailored to explain physical reality, stating an identity across frameworks.
2. The scientific method and category mistakes: data vs. interpretation as a fundamental problem
For example, the quote says "decisions", "subjective experiences" and "covert attitudes" can be detected. Following Reynaert, this is simply not true: neuronal activity can be detected (data) and "decisions", "subjective experiences" and "covert attitudes" are voluntarily and consciously identified (interpreted as being identical) with the physical events detected.
Basically, the neuroscientists that say they can measure mental states have decided that the physical data is the mental state before measuring anything at all since all they are able measure is the occurrence of physical events and the time difference between these occurrences. The identity or non-identity of a physical and a mental event is part of their theory. But the basic hypothesis as such - the identity of neural activity and mental states - is probably not testable at all and cannot become part of the scientific endeavour proper, i.e. it is a contingent premise that determines the interpretational outcomes of the studies.
Thus, we here have a perfect example of a genuinely philosophical question about whether there is an identity or not.
3. What about the dualists? How do they explain the outcomes?
In some sense, dualists do not have to "explain" anything there, they can simply point out that this interpretation is nonsensical (as Reynaert does at length - it is about this being absurd) and that there is no discrepancy to be explained as we are talking about totally different things here.
That being said, substance dualism as an ontological position obviously runs into different problems, i.e. mind-body interaction. It can simply be stated that it works so-and-so, but whatever the explanation ends up to be, it can hardly be proven as the correct interpretation. All there can be is more or less coherent justifications and arguments: It is speculative metaphysics.
Nevertheless, there are many variants of explanations out there that try to explain the temporal discrepancy between physical and mental events that somehow seem to have something to do with each other, though. One possible way to think here is given by Daniel Dennett in Freedom Evolves, p.229:
When you think you're deciding, you're actually just passively watching
a sort of delayed internal videotape (the ominous 300-millisecond delay)
of the real deciding that happened unconsciously in your brain quite a while
before "it occurred to you" to flick.
In other words: You could theorise that there actually is an unconscious decision and that this one's "yours" in a meaningful sense but you perceive it much later on and only attribute/interpret it as your conscious decision happening in that moment.