I actually spent some time studying category theory this year so I feel I can provide a little more insight here than has been offered yet.
I have seen at least one research article in which it is succinctly expressed, “topos theory is basically modern logic”.
A topos is a specific, very technically defined mathematical object, construct, or structure (whatever you want to call it).
Having spent a few months this year studying category theory, I have come to feel pretty strongly, you cannot do mathematics justice through approximations. Many people have heard a layperson’s simplification of topics like Gödel’s incompleteness theorem, or quantum mechanics, or quantum computers, or blockchains, or string theory, and the issue is that if you do not study it authentically - having studied the actual mathematics in which the ideas are expressed - you pretty much never succeed in actually understanding it authentically either. The problem with this is that it leads to misunderstanding - inaccurate belief - which charades as comprehension. My entire life, I have felt like I kind of get the idea of Gödel’s incompleteness theorem, but as I have learned more logic slowly, I have often had moments of revelation where I realize, “I basically never had a clue what the theorem is even saying, let alone why anyone would think it is interesting, let alone very interesting”. (I am still in that process, which maybe hints that this still happens to people as they continue to study something more.)
Trying to explain category theory (which can be seen as the general field that topos theory comes from or lies within) in terms of loose generalizations simply cannot get you anywhere beyond vague-sounding truisms - “it’s a kind of mathematics of structure, which shows the structure of other mathematical objects, and their relationships to one another…” It doesn’t sound like a particularly illuminating idea.
I will be honest with you, through my exposure, I have come to feel that category theory may be the most extraordinary field of knowledge I have encountered in my life. It is hard to convey how amazing it is. And I know many people who get exposed to it feel the same way. It is profound.
Anyway. Maybe I can actually try to explain a topos in technical - yet clear - terms (I’ll come back and edit this answer). But a short answer to your question for how, is…:
Category theory is one of the most general fields of mathematics imaginable, because it starts with a very simple, basic structure (a category), and shows how many things in the world of math can fit into that structure. It then builds on itself by mind-blowing applications of the idea of “abstracting something into its structure” to itself: you can have a category of categories, for example. This is a category whose objects are also categories. Thus, you can study the relationship of different categories to each other, using a category. Also, there are many, many specific properties a given category can have, and this leads to acquiring a very large and esoteric-sounding vocabulary to express the relationship between all these properties - things like ‘functors’, ‘monads’, ‘natural equivalences’, ‘fibrations’, ‘quasicategories’, ‘dagger categories’, ‘infinity categories’, and so on.
Basically, a topos is one such special type of category, with certain special properties. Some people see it as having come to define the modern field of logic, because it provides a way to specify the unique properties of many, many different kinds of logic (yes, there are different kinds).
Basically, there is this finding that people find to be extremely deep - they call it “the computational trilogy”: https://ncatlab.org/nlab/show/computational+trilogy
I’ll leave you to read it; but the idea is that three very different ways of thinking about the mathematical world can be shown to be structurally identical, in a way: computation - logic - and structures.
I found category theory very hard to study. I would recommend you start with some simple books, for example by David Spivak, and join the chat group that some people from the nCatLab site run.
As I said, as you adapt to this mathematical way of thinking, one ceases to use “hands-wavy” verbal approximations or interpretations and becomes increasingly aware of the usefulness, compactness, economy, and necessary specificity of mathematical definitions.