A preposition appears to be needed when that is used in sentences such as:
That's the store where I bought my computer.
That's the store that I bought my computer (at?).
with exceptions occurring, per page 483 of Oxford Practical English Usage, for "somewhere, anywhere, everywhere, nowhere, and place."
That's the place where I bought my computer.
That's the place that I bought my computer (at?).
A comment on a related question at ELL states:
In practice, most native speakers wouldn't get bogged down in prepositions and relative pronouns here - we'd just say "We need a place to stay". But for no specific reason that I can identify, if I replace place with house there it seems to require the preposition: "We need a house to stay in".
I agree with the notion that the preposition is required following that. What I can't do is wrap my head around exactly why that should be.
What difference is there between where and that in these sentences that calls for the omission of a preposition when using where and the inclusion of a preposition when using that?