They used "at" because that's the preposition we use before giving an address.
I live at 24 Macadamia Avenue.
They turned up at 17, High Street, Meopham.
The Prime Minister lives at 10 Downing Street.
Several things have happened since she took over at 10 Downing Street.
In the original piece, "10 Downing Street" is shortened to "No 10" - all Brits know what address that refers to.
It wouldn't be ungrammatical without "at" but it would have a very different meaning! She didn't take over the building! She moved into it when she took over the job of Prime Minister.
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