With respect to the following link: http://www.archlinux.org/news/libpnglibtiff-rebuilds-move-from-testing/
Could someone explain to me why a program should be rebuilt after one of its libraries has been updated?
How does that make any sense since the "main" file is not changed at all?
If the signatures of the functions involved haven't changed, then "rebuilding" the program means that the object files must be linked again. You shouldn't need to compile them again.
An API is contract that describes the interface to the public functions in a library. When the compiler generates code, it needs to know what type of variables to pass to each function, and in what order. It also needs to know the return type, so it knows the size and format of the data that will be returned from the function. When your code is compiled, the address of a library function may be represented as "start of the library, plus 140 bytes." The compiler doesn't know the absolute address, so it simply specifies an offset from the beginning of the library.
But within the library, the contents (that is, the implementations) of the functions may change. When that happens, the length of the code may change, so the addresses of the functions may shift. It's the job of the linker to understand where the entry points of each function reside, and to fill those addresses into the object code to create the executable.
On the other hand, if the data structures in the library have changed and the library requires the callers to manage memory (a bad practice, but unfortunately common), then you will need to recompile the code so it can account for the changes. For example, if your code uses malloc(sizeof(dataStructure)) to allocate memory for a library data structure that's doubled in size, you need to recompile your code because sizeof(dataStructure) will have a larger value.