Learn Romanian: Casa lor e de două ori mai mare decât a noastră.

In Romanian, you do not use an "as big as" structure for this kind of multiplier. Instead, you literally say "two times bigger than" (de două ori mai mare decât). This is the standard way to express proportional comparisons.

How exactly does de două ori translate to "twice"?

It literally means "of two times". The word ori is the plural of oară, which means "time" in the sense of an instance or occurrence. To say "three times" or "four times", you would just swap the number: de trei ori, de patru ori.

Why is there an a before noastră at the end of the sentence?

The a is a possessive article that replaces the noun casa so you do not have to repeat it. It acts like the English word "ours". Because casă is a feminine singular noun, you must use the feminine singular possessive article a.

Does the word lor change depending on the gender of the house?

No, lor ("their") is invariable. Unlike most other Romanian possessives (like meu or noastră), which change to match the noun's gender and number, lor always stays exactly the same.