Well, basically it's a fancy way of saying "a lot," yes.
If you want a longer (and possibly less accurate) answer:
I'd take "to a large extent" to include the idea of proportion, percentage... a large slice of the economy depends on tourism.
Let me try to explain:
10.000 people is "a lot" of people, but it's not a large part of 6 billion people...
So tourism could be very important, but in this case it's probably also very important when compared to other industries and whatever the economy of a country is made of.
yes, you are correct, "extent" means "how much of something", and the word in front of it tells you how much, "great" means "a lot". So in the context, it means that the tourist industry is very important to the country's economy.
Oh, and you may also see "lesser extent", which means "not a lot" or "not as much as something else."