Although Freud made the unconscious famous the idea that moods, feelings, desires and beliefs can be unconscious even in humans is pre-philosophical and unmysterious as observations of sleep, dreams, absent-mindedness, obliviousness, and colloquialisms like "unaware of her feelings", "beside himself with anger" or "doesn't know himself" attest to. Aside from introspection, what puts these folk mental states in play is their utility in conceptualizing behavior of other humans and animals, or even some things. There is as little reason to assume that they cease unless consciously attended to as to assume that desks and chairs blink out of existence when we turn away, so awareness was never definitive of them. Of course, there are also high minded Mental States of introspection imbued with qualia, superknowledge and other supposed perfections.
But one need only adopt a broadly functionalist approach to mind to work with refined relatives of folk mental states without any complications. This includes not only psychoanalysts, but even transcendental idealists like Kant and Husserl, who infer conditions of possibility from appearances. For example, Kant informs us in the first Critique that the famous "productive imagination" responsible for all our cognitive syntheses "without which no knowledge will be possible at all", is "blind", and "we are rarely conscious of it". And materialists, who imagine mental states and consciousness to be manifestations of some physical processes, have no reason to suggest that one can not manifest without the other. By the way, Freud in his youth came up with a Pavlovian research programme in psychology, which he abandoned only because experimental methods available at the time were inadequate. Even Plato, a rationalist par excellence, writes about the unconscious knowledge we bring forth in anamnesis, and describes the Chariot of Soul driven by the horses of reason and passions, with passions not always consciously attended to. It would take a special kind of rationalist stuck in the Cartesian theater to insist that folk mental states must be conscious at all times.
What of the Mental States? Well, their continuity with unconscious folk states in humans should at least give one pause about their exceptionality, do we believe that fear or jealousy radically transform once one becomes aware of them? So should neuroscience experiments suggesting that some "superknown" introspections are confabulations after the fact that people can not always time accurately. As Roskies describes, "some aspects of awareness of agency seem constructed retrospectively. A recent study shows that people’s judgments about the time of formation of intention to move can be altered by time-shifting sensory feedback, leading to the suggestion that awareness of intention is inferred at least in part from responses, rather than directly perceived". Many philosophers and psychologists now conclude, as Schopenhauer and Wittgenstein once did, that consciousness is not essential to exercise of free will either, e.g. Rosenthal writes that "nonconscious volitions might occur simultaneously with neural initiating events and might even be identical with such neural events. At most, Libet's work shows not that volitions do not initiate actions, but that conscious volitions do not", and Levy goes further saying that "decisions, volitions and the formations of actions must all ultimately occur unconsciously". See Mele's review Recent Work on Free Will and Science.
It seems to me that it is the proponents of Mental States that need to do the lion's share of explaining.