I don't even know how to properly ask this, but how can one prove things happen without them knowing?
Things only exist for me when I'm aware of them, either by direct contact (I see it, I feel it etc.) or indirect contact (I was told by someone). But if that's the case, how one can be sure they're not "the center of the universe". How do I know things I don't know exist?
You raise two issues. Solipsism, the question of whether any other mind truly has subjectivity, or only appears to given that you can't directly verify like you do your own. And the consistency of unperceived phenomena.
Solipsism is a classic question for philosophy. Descartes approach was to apply successive degrees of skepticism, at the most extreme recognising only that we can be sure we are thinking, because contemplating the question in any way verifies that, to noting the clear and distinct quality of this understanding is a kind of evidence that is itself a useful criteria. And then building a hierarchy of knowledge from there. In modern philosophy philosophical zombies are used as a thought experiment to understand internal experience and qualia and whether they are essential for minds, or instead are epiphenomena. So it useful to consider solipsism, and philosophical work can be done from considering it. But someone truly embracing solipsism as a stance shouldn't be interested in other people's views, and so can't meaningfully engage with philosophy. There is more evidence for an external world, than against it, as well as practical arguments that it is useful, and beneficial to gene replication to believe in it - and our conditioning by genes is a crucial part of explaining how we can ask questions at all.
Unperceived phenomena are directly addressed by the thought experiment of whether a tree falling with no one to hear makes a sound. I would describe the physics perspective as based on conservation laws, and continuous symmetries which are proven equivalent. Known in enough detail, the future states of things predicts, or constrains, outcomes. It would be non-sensical to describe what seem to be universal laws describing the cosmos, as being founded in a local perspective. They cannot be altered by a persons mental activity, so they are external, and define the consistencies we call the external world. We can use complex systems on the edge of chaotic behaviour, and apply statistics to the outcomes, to show they have continued to evolve while not being viewed, in a way that could not be simulated except by running the system or something directly equivalent to it. What will be compelling will depend on exact reasoning to get to and flavour of solipsism though.
Intersubjectivity, is useful for understanding how private experiences can be shared. In Yogacara Buddhist thought, also called the mind-only school, the universe is considered to be composed of narrative chains, that causality occurs within these, rather than among mind-independent material. This kind of pansychism still has to explain the regularities we experience of the world, and the metaphor of Indra's Net can help us understand this: as an intersubjective space occuring at the interface between mind and world mutually arising. We can never access phenomena not processed by someone's subjectivity, yet we find evidence for regularities unaffected by subjective processes. Our minds and the world arise and develop together, and become more complex through interaction, especially between minds, through sharing perspectives, intersubjectivity. This helps us understand how meanings can arise that are shared across minds like our own, yet cannot be understood as truly 'out there', or as resulting from a Private Language or from a priori experiences. More here: Is the idea of a causal chain physical (or even scientific)?
If a solipsist accepts the validity of some version of the PSR (Principle of Sufficient Reason), they can prove the existence of something outside of their world of representations. Events within the world of representations must obey a causal principle, according to which every event within some specifiable category has a causal explanation. The totality of mental representations also has to have some causal explanation. There are two options here. (1) The causal explanation appeals to the nature of the solipsistic consciousness itself or (2) The causal explanation appeals to something outside of the solipsistic consciousness. Suppose it is (1). This alternative is internally incoherent. Here's how to see why. The solipsistic consciousness X potentially has access to a very complex set of evidence of other experiencing subjects with almost qualitatively indistinguishable experiences. Some of these apparent experiencing subjects would produce verbal reports that contradict other self-representations of the solipsistic consciousness X, e.g. that X has only been alive for 30 years, the world has existed for billions of years, regularly talks to other independent subjects whose existence does not depend on X, etc. The solipsistic consciousness has to interpret these verbal reports as just bundles of representations that are ultimately its own. There are no independent subjects of experience behind these bundles of representations. But then X has contradictory representations. This won't be problematic if one could introduce a distinction between truth and falsehood relative to how the world really is (one sets of representations measures up to how reality is, another fails, and so on). But such a distinction is not available to a solipsistic consciousness since the latter's representations are not answerable to anything outside of the solipsistic world. X cannot say to itself: "that bundle of representations associated with apparent subjects A, B, C is false and, thereby, resolve, the internal contradiction." All of these mutually inconsistent representations are self-generated and X cannot have any rational basis for distinguishing between them. The only source of any resulting consistency, by privileging the solipsistic belief against the other contradictory self-generated representations, can be something a-rational like a decision. In other words, a solipsist cannot self-consciously engage in rational processes. So rather than ask how one can prove there is something existing unperceived, one should ask how one can ever have anything like a real solipsistic belief (and the emphasis here is on belief because beliefs are by their nature mental states subject to rational evaluation and carry along the distinction between truth and falsehood with themselves). Al alternative formulation of the same point is that option (1) above ultimately leads to the conclusion that no real causal explanations within a solipsistic world are possible or any causal explanations of the totality of solipsistic representations. To have a representation of a world of objects stably obeying law-like regularities, one's representations have to be internally coherent at a minimum. Therefore, option (1) is incoherent. There has to be something outside of a solipsistic consciousness to causally explain both the relations among its representations and the totality of its representations.
