How does the size of the prime affect Elliptic Curve Bit Security? I am using the MIRACL Library to implement an Elliptic Curve Diffie Hellman based Key Exchange according to ECDH-Scheme-Wikipedia.

Referring to the Miracl Docs they suggest a few curves. 
Each curve is with respect to a prime p which is n bits in length Miracl Docs EC suggestions. 
How does the n bit length of the prime e.g. from the ssc-192 suggestions refer to the provided bit-security?
Elliptic curve security relies on the hardness of discrete logarithm on that curve. (Well, that's a simplification, but this will do for this answer.) When the curve contains N points, it takes an effort of roughly sqrt(N) "elementary operations" to break discrete logarithm.

A prime p of "k bits" means that p is less than 2k, but greater than 2k-1. The number of points on a curve defined over the field of integers modulo p will be close to p (that's Hasse's theorem). Thus, if the prime p has length k bits, the breaking effort will be on the order of the square root of 2k operations, i.e. about 2k/2. It is usually considered that the current, absolute barrier on that which is technologically feasible is between 280 and 2100, so a safe curve would need a prime size of at least 160 to 200 bits. The "ssc-192" curve falls in that domain, so it could be deemed "probably secure".

Cryptographers are fashion victims, and thus always want powers of 2; thus, they will usually ask for "128-bit security" which then translates to a 256-bit prime p.

See this site for a lot of pointers to information on the concept of key strength.