It's actually simple: you need inline when you want to write a definition (of a function or variable (since c++17)) in a header. Otherwise you would violate odr as soon as your header is included in more than 1 tu. That's it. That's all there is to it.
Of note is that some entities are implicitly declared inline like:
- methods defined inside the body of the class
- template functions and variables
- constexpr functions and variables
Now the question becomes why and when would someone want to write definitions in the header instead of separating declarations in headers and definitions in source code files. There are advantages and disadvantages to this approach. Here are some to consider:
optimization
Having the definition in a source file means that the code of the function is baked into the tu binary. It cannot be inlined at the calling site outside of the tu that defines it. Having it in a header means that the compiler can inline it everywhere it sees fit. Or it can generate different code for the function depending on the context where it is called. The same can be achieved with lto within an executable or library, but for libraries the only option for enabling this optimization is having the definitions in the header.
library distribution
Besides enabling more optimizations in a library, having a header only library (when it's possible) means an easier way to distribute that library. All the user has to do is download the headers folder and add it to the include path of his/her project. In the case of non header only library things become more complicated. Because you can't mix and match binaries compiled by different compiler and even by the same compiler but with different flags. So you either have to distribute your library with the full source code along with a build tool or have the library compiled in many formats (cpu architecture/OS/compiler/compiler flags combinations)
human preference
Having to write the code once is considered by some (me included) an advantage: both from code documentation perspective and from a maintenance perspective. Others consider separating declaration from definitions is better. One argument is that it achieves separation of interface vs implementation but that is just not the case: in a header you need to have private member declarations even if those aren't part of the interface.
compile time performance
Having all the code in header means duplicating it in every tu. This is a real problem when it comes to compilation time. Heavy header C++ projects are notorious for slow compilation times. It also means that a modification of a function definition would trigger the recompilation of all the tu that include it, as opposed to just 1 tu in the case of definition in source code. Precompiled headers try to solve this problem but the solutions are not portable and have problems of their own.