For example, suppose that you have a bug in the body of a function, but did not provide a type annotation for it. The function might still compile but not with the type you want. The compiler will only notice something is amiss when you try to call the function and it turns out that the function's inferred type doesn't fit the call site.
Basically, global type inference in the absence of type annotations means that changes in one part of the file can affect inferred types very far away. In practice it's best to use type annotations to limit inference to small sections, so that type errors are reported close to what caused them.
The problems are:
1. Type errors become much less localised and much harder to understand. Instead of "the function you're editing is supposed to return a string but it doesn't" you get "this other function three stack frames away that indirectly calls the function you're editing can't pass some value into println".
2. The inferred types are as generic as possible, which also means they are as hard to understand as possible.
It's infuriating when a type error can "jump" across global functions just because you weren't clear about what types those functions should have had, even if those types are very abstract. So early adopters learned to sprinkle in type annotations at certain points until they discovered that the top-level was a good place. In OCaml this pain is somewhat lessened when you use module interface files, but without that... it's pain.