Short Answer
How to imagine reality outside of space and time?
As others have noted, it seems counterintuitive and paradoxical to have a reality outside of spacetime. The roots of the concept of reality are moored in the senses being able to make sense of mass, space, and time, etc. So, to do so, or at least to understand trying and doing so really requires an examination of imagination. Thus, the question reduces to in what ways can one imagine a reality outside of spacetime. Different traditions have different blends but use various aspects of imagination such as visualization, storytelling, and argumentation.
Long Answer
Imagination has several aspects. One can use the imagination for the senses such as iconography and mental imagery, and one can use imagination to create language, narrative, and justification. Generally, if one attempts to defend the idea of a reality outside of spacetime, all of these may be part of the psychological processes that are used. Theists in particular, who often defend realities outside of our sensory reality are a perfect example since they build edifices, adorn them with pictures, encourage prayer and meditation, have cosmogenies, stories, and doctrines.
It would be possible to write a book on traditions used to construct belief in realities outside of realities, but as this is a philosophical forum, let's focus on epistemological methods. There are two grand, philosophical traditions and one of more recent origins. Both Platonism and Christian theology argue the existence of realities outside of spacetime, and more recently and trendy is the notion of modal realism and multiverses. All three of these schools of thinking posit at least one reality outside of our shared physical reality with its spatial extension and causal nexus. From WP:
The physics community has debated the various multiverse theories over time. Prominent physicists are divided about whether any other universes exist outside of our own.
Some physicists say the multiverse is not a legitimate topic of scientific inquiry.4 Concerns have been raised about whether attempts to exempt the multiverse from experimental verification could erode public confidence in science and ultimately damage the study of fundamental physics.5 Some have argued that the multiverse is a philosophical notion rather than a scientific hypothesis because it cannot be empirically falsified. The ability to disprove a theory by means of scientific experiment is a critical criterion of the accepted scientific method.6 Paul Steinhardt has famously argued that no experiment can rule out a theory if the theory provides for all possible outcomes.7
It would be a long response that attempted to show how each of these three schools defends their beliefs, so I'll just give a quick run down on Platonism. Platonists hold roughly that the circles you and I draw and think about are some sort of reflection of a perfect circle that is an example of a Form.
A theory of substantial forms asserts that forms (or ideas) organize matter and make it intelligible. Substantial forms are the source of properties, order, unity, identity, and information about objects.
The concept of substantial forms dominates ancient Greek philosophy and medieval philosophy, but has fallen out of favour in modern philosophy.1 The idea of substantial forms has been abandoned for a mechanical, or "bottom-up" theory of organization.2 However, such mechanistic treatments have been criticized for the same reasons atomism has received criticism, viz., for merely denying the existence of certain kinds of substantial forms in favor of others (here, that of atoms, which are then thought to be arranged into things possessing accidental forms) and not denying substantial forms as such, an impossible move.
This notion is frequently based on an argument that since you and I can both objectively know something like an abstract mathematical object, then it must exist in some sense similar to rocks and stones. (Caveat: my personal views come from a naturalized epistemology and these sorts of claims are woo.) From there neo-Platonists engage in a variety of epistemological justifications that revolve around claims focused on ontological dependence. For instance, an abstract mathematical object has an existence independent of human thought, because it is discovered. Since it is objective, then it must, like empirical objects must exist in a realm. Since the realm isn't directly accessible to our senses, our minds have some other avenue for detecting them. And so on.
Now, if you're like me and a proponent of scientism, you don't see much of a difference between the Realm of Forms and Vallahala and alternative realities. But remember that for many people, there are many forms of evidence besides empirical and rational evidence. Some people are content to place their first principles in articles of faith. Others, as in the comments, have suggested that some people can reject scientific realism outright. And many may reject a rational basis for evidence electing to reject the pursuit of epistemology entirely claiming that divine revelation trumps human argumentation.
So, the defense of realities outside of our shared spacetime come in too many flavors to list, but they tend to have much in common. They often use cosmogenies, doctrines, and narratives that create a holistic, functioning worldview that is often promulgated in communities of belief around the world for centuries or even millennia. If you have a hard time imagining, I suspect it might have to do with the fact that you are born outside of a fervent form of these communities which tend to push theories of the world that are laden with values the community espouses. There are many views in the philosophy of social science that suggest reality is largely a social construction.
