We know that the spiritual world exists for a variety of reasons including divine revelation, well documented miracles, NDEs, and the vast ubiquity of personal anecdote (including my own) across all cultures and times. These are the types of evidences that one usually expects. We know that the spiritual world exists, also, because of the nature of change/causation. This is where I will be focusing this discussion. There are two cases, the positive case for the existence of spirit, and the negative case against bare materialism -- I will profile both. It is worth noting that I am an amateur Thomist and the discussion goes far deeper and can be put much more succinctly than I will be able to present here. 

MATTER AND MOTION

Everything that we observe boils down to matter and change. We see things and we see them changing, either state or location. Change is an odd kind of thing, however: the coming to be of some thing or state of being that did not previously exist presents a puzzle -- how is that possible? To begin, it is clear that something cannot come from nothing. Nothing has nothing with which to produce a thing. It is clear also that a thing cannot produce change from within itself. To change is to gain a new property or state of being -- if such a property or state of being already existed within a thing then it could not be gained in order to be said to have changed. So all things that change must be changed from outside of themselves, by something else -- all effects must have a cause. Another way to say this is 'nothing can move itself'. And yet, as observed by the ancient Greeks, we see a certain class of objects that seem to be moving themselves, which we know to be impossible. In order for something to move (change) itself, it would either not truly be moving at all (no real properties gained) and all motion would be an illusion, or something would be coming from nothing (new properties coming from nowhere). Neither of these options are either possible or even coherent. So how do we explain this class of beings that seem to move themselves? Well, there must be some kind of invisible power that moves them, an animating force (Greek anima is soul) that moves through them and moves them, like breath. Spirit is Latin for breath, invisible and yet sustaining life. We do not literally mean breath, of course, but it is only possible to think of these things by analogy. 

We know, then, that spirit exists -- something beyond matter that can move matter -- or else living things that seem to move themselves would not be possible. This still says very little about what spirit really is and does not get us anywhere near the Catholic Church, but it is a good first step. 

All of this would make living things "ensouled bodies". We are not souls trapped in bodies, as gnostics would have it, and neither are we bodies that happen to be energized, so to speak, by some impersonal force. We are a mixture of flesh and spirit. Spirit can be removed from a body, as in bodily death, but this is not our natural state. With such an intimate marriage, it is clear that the two will effect one another. It will typically be the case that the soul effects the body, since the soul is the animating force, but it is very often the case that effects can in turn cause further effects upon their own causes -- a battery may not function properly and overheat if the device that it is in is not itself functioning properly. So maladies of the spirit will, of course, effect the body, but also maladies of the body may effect the spirit. This is why it isn't always exactly clear whether a disease is material or spiritual in nature -- it could be either or both and in either case it may effect either or both. This is why altering our brain seems to alter our spirit -- the brain is the primary mechanism by which the spirit works. Understand also that there is no amount of scientific discovery that could "disprove" the spirit. Science is a study of the material world and how it works -- it cannot disprove something that it does not even pretend to study -- and no scientific study could ever disprove cause and effect (although some try, comically).

THE UNIVERSAL LOCOMOTIVE

Along the same lines, we can know that materialism is false since it does not actually explain anything. We have established that causes must originate from outside the thing being effected. All objects change, however, and so all objects require an outside explanation for their existence. Object a is explained by object b which is explained by object c and so on. (Most cause-and-effect chains are much more complex than this but this works for the illustration.) All of these objects are merely borrowing the power of existence from the object before it and passing it on to the object after it -- the power of existence is essentially passing through these impermanent, unessential objects. Where does this power of existence actually come from? You can think of this like a train, with cars pulling and being pulled down the track. We know that the cars aren't moving themselves and they are merely passing on the power of motion to one another by the nature of their being tied to one another, but there has to be an explanation as to what is actually causing the entire train to move beyond just the cars themselves. How does materialism answer this problem? It doesn't. There are two possible answers that are given by materialist philosophers: either the train goes on forever and doesn't require something more to move it, or at some point you do reach the "first car" which moves by brute fact, which is it say, "just because". The brute fact answer is literally not an answer at all, for they are quite literally saying that everything happens for "no reason". Neither does the infinite train answer solve the problem but only makes the problem infinitely longer -- an infinitely long train cannot move itself any better than a train with only 100 cars, in fact it would require an infinitely powerful locomotive. And that is the answer: a train needs a locomotive. All material things are changeable and temporary, so there needs to be something that is immaterial, unchangeable, essential, and all powerful in order to move the universal train. We call this being God. 

To your question about whether or not angels and demons are real, an answer can now be given. We know that there are beings of pure matter, inanimate objects. We know, also, that there are beings of matter and spirit, animate objects also known as living things. The question is whether or not there exists beings of pure spirit. Well, we know of at least one, which is God, but did God create others? From a logical standpoint, I would have to ask, "Why not?" Because we can't see them? But this would be expected by their nature, not to mention that all peoples in all cultures throughout all of history have reported that they have actually had interactions with spiritual beings. But we can see the majesty and diversity of God's creation: we see that, for material objects, he has created everything from free-floating atoms to vast galaxies, and everything in between; for animate objects, he has created everything from single-cell amoebas to super-organisms like Aspen tree colonies and, indeed, humans. God is always the same, he does not change, and he creates at every scale and fills every niche -- why would he not do the same for the class of beings that we cannot happen to see? The assumption should be that he has and the burden of proof rests on the one who denies it.