See the Rationale section from PEP 424 that introduced __length_hint__ and offers insight on the motivation:
Being able to pre-allocate lists based on the expected size, as estimated by
__length_hint__, can be a significant optimization. CPython has been observed to run some code faster than PyPy, purely because of this optimization being present.
In addition to that, the documentation for object.__length_hint__ verifies the fact that this is purely an optimization feature:
Called to implement
operator.length_hint(). Should return an estimated length for the object (which may be greater or less than the actual length). The length must be an integer>= 0. This method is purely an optimization and is never required for correctness.
So __length_hint__ is here because it can result in some nice optimizations.
PyObject_LengthHint, first tries to get a value from object.__len__ (if it is defined) and then tries to see if object.__length_hint__ is available. If neither is there, it returns a default value of 8 for lists.
listextend, which is called from list_init as Eli stated in his answer, was modified according to this PEP to offer this optimization for anything that defines either a __len__ or a __length_hint__.
list isn’t the only one that benefits from this, of course, bytes objects do:
>>> bytes(Foo())
len
getitem 0
...
b'\x00\x01\x04\t\x10\x19'
so do bytearray objects but, only when you extend them:
>>> bytearray().extend(Foo())
len
getitem 0
...
and tuple objects which create an intermediary sequence to populate themselves:
>>> tuple(Foo())
len
getitem 0
...
(0, 1, 4, 9, 16, 25)
If anybody is wandering why exactly 'iter' is printed before 'len' in class Bar and not after as happens with class Foo:
This is because if the object in hand defines an __iter__ Python will first call it to get the iterator, thereby running the print('iter') too. The same doesn’t happen if it falls back to using __getitem__.