Choice B changes the intended meaning.
original sentence communicates the sense of increase in the measure - numbers - the number has increased.

E.g. now there are 10 gyrfalcon's numbers when earlier there were 3 gyrfalcon's numbers.

Use 'greater than' when DESCRIBING NUMBERS
We normally say - "Value x is 5 times greater than Value 5" and not "Value x
is 5 Times more than Value y". Here X represents a number (density of wolf,
number of birds etc)

Example:
A wildlife expert predicts that the reintroduction of the caribou into northern
Minnesota will fail if the density of the timber wolf population in that region is
greater than one wolf for every 39 square miles.

Generally the countable things use GREATER THAN

Use MORE to modify the THING BEING COUNTED:
There are ten MORE BIRDS in Park A than in Park B.

Use GREATER to refer to the VALUE of the thing being counted:
The NUMBER of birds in Park A is GREATER than the NUMBER of birds in Park B.

The gyrfalcon, an Arctic bird of prey, has survived a close brush with extinction; its numbers are now five times greater than when the use of DDT was sharply restricted in the early 1970s.

A) extinction; its numbers are now five times greater than
B) extinction; its numbers are now five times more than
C) extinction, their numbers are now fivefold what they were
D) extinction, now with fivefold the numbers they had
E) extinction, now with numbers five times greater than

I chose B. The correct answer is A. I'm confused between 'five times greater than', 'five times more than' and 'fivefold the numbers'...when do we use each one of them?

Many thanks!
Lydia

Lets try to see what's wrong with each incorrect answer.

The gyrfalcon -> Singular subject ->Eliminate C and D.
Answer choice E is incorrect because the usage of now changes the original meanings as if they survived this period and now with such a large number they are causing XYZ. I mean the word now points to a result or a causal relation.

Between A and B. A is correct usage. In English we always say number 5 is greater than 3 and not more than 3.
Hence A wins.