Numbers Theory is the discovery of rules and order among integers. It is focused on divisibility and primality.
It is useful for cryptography because it allows one to devise intractable computational tasks that can be made easy using a known 'key' -- the paradigm of ciphers.
Number theory is a rich source for one-way functions: computations that are easy one way, and difficult in reverse.
For example: it is easy to multiply x=9929 by y=9931 (both primes) to xy= 98,604,899, but given xy, it is quite difficult to find x and y.
Number theory is the only mathematical foundation on which asymmetric ciphers have been constructed, allowing cryptography between strangers (necessary for e-commerce).