The obvious answer is that olives grow really well in Italy. Beyond that, olive oil is nutritious, useful, tasty and it doesn't rot. For an empire built around extensive (but slow) trade networks, it was the perfect product.

And yes, it was massively popular outside of Rome. Olive oil was shipped all over the empire and beyond. Aside from being a food staple, it was even a "prestige item" not unlike wine is today. Actually, not unlike olive oil is today. There were everyday table oils and then there were the fancier varieties that would have been more expensive.

As a tangentially related question on the topic of trade and cooking oil, was there similar demand and localized production of butter in ancient central/northern Europe? Were there any regions that specialized in making and trading butter?