The Grandfather Clause made it so that if your grandfather could vote, so could you. The Grandfather Clause was established along with the literacy tests, poll taxes, etc. There were two ways to vote- either your grandfather voted (which most whites did have and most blacks did not have), or you passed the literacy tests and payed the taxes. The clause made it so most whites were able to bypass the tests while the blacks had to take the tests, which were actually impossible to pass.
These people were probably called pre-1867 voters because the 13th, 14th, and 15th amendments were made just before 1867, making these clauses and tests illegal.