Based on mundane world examples of broken English, the likely explanation for it starting is that the House Elves have, or had, their own language, and English was basically grafted onto it. It's been said that as long as you can communicate, broken language may be less aesthetically pleasing, but it's not wrong. In early Human / House Elf interaction, having any sort of common language would be a benefit.
As to why it's persisted, this often happens in the mundane world as a form of cultural identity, with some pidgins or creoles becoming their own dialect, c.f. Black Vernacular, Jamaican English, or Louisiana Creole. The usage of non-standard (to use) English is a way of establishing that you are indeed separate from the culture you live in. A somewhat uglier interpretation is that the Wizards prefer the Elves to speak in this Broken English because it clearly delineates the difference in society and, in their mind, the superiority of Wizards, justifying the slave labor, and the Elves go along with it because it smooths their way to speak "correctly" according to what their master wants.
Taking this a step further, Rowling probably used the Broken English trope (warning, TV Tropes link) to emphasize the power inequality between Wizards and their House Elves in the same way that prior books have given Black or Asian sidekicks agrammatic English.