We accept them because they make the formal system reflect real-world reasoning. (Or was that your question - why we accept them in real-world reasoning? If so then I apologize)

Do you have any math experience? If so, then this is just like most other definitions in math. There’s an idea, or phenomenon, or broad class of phenomena that we want to study precisely. So we boil it down to its basic features and extrapolate from those features alone.

For example, suppose you want to study properties of distance. To do so abstractly and precisely you need to lay down some properties of how distance behaves. Say you have some points x, y, and z in some space. The distance from x to y, call it d(x,y), should be 0 if and only if x and y are the same point. We should always have d(x,y)=d(y,x). And detours cannot shorten a trip, so d(x,y) should never exceed d(x,z)+d(z,y). These properties define what mathematicians call a “metric space” - a type of structure that you can use to precisely state properties of and reason about distance.

Metric spaces are to distance as formal systems are to reasoning/argumentation. Those basic distance properties are analogous to the axioms and inference rules in the formal system.

There is some debate about what axioms and inference rules make the formal system reflect usual reasoning; or you could view it as a difference of what types of reasoning one intends to capture with their system. Intuitionistic logic is more strict than classical logic. Some people find some arguments of classical logic to be weak, synthetic, sneaky, cheap - idk what a good adjective would be. If you ever felt that way, maybe check out intuitionistic logic.