My house is (2x) twice as big as yours.
My house is (⅓) a third as big as yours.
My house is (½) half as big as yours.
My house is (1⅓x) a third again as big as yours.
My house is (1½x) half again as big as yours.
The use of again after a fractional expression is used to represent that something is of equal size plus the fractional number.
This is the sense of again that's being used:
4 : in addition : BESIDES
// Again, there is another matter to consider.
The use of again can come after any fractional representation.
Outside of the particular phrasing in this context, you might hear something like this:
Give me a pint, and then half a pint again.
To paraphrase and analyze the context of its use in house sizes, it means the following:
My house is [the same size] as yours and [this fraction] bigger [in addition].
The use of half again specifically is common enough that it's actually become an idiom:
half again as much/many as
—used to say that the size or amount of one thing is 50 percent more than another
// If 100 people were expected and 150 came, half again as many people came as were expected.
// This dress costs half again as much as that one.
Note that the definition includes US, so this could be less common in the UK.