prime numbers explanation

This is the Bertrand's postulate: For every n > 1 there is always at least one prime p such that n < p < 2n.
The proof is complicated. It has no significance as far as GMAT is concerned. You are not expected to know Bertrand's postulate and this is not an intuitive number property.

The second property you mentioned is important. This is how you can explain it.

Every integer equal to or greater than 6 will be of one of the 6 forms: 6n or 6n+1 or 6n+2 or 6n+3 or 6n+4 or 6n+5

Note here that 6n is divisible by 6.
6n+2 = 2(3n+1) i.e. divisible by 2
6n+3 = 3(2n+1) i.e. divisible by 3
6n+4 = 2(3n+2) i.e. divisible by 2

Hence none of these can be prime. So a prime must be of the form 6n+1 or 6n+5 (which is the same as 6n-1)

Notice that 5 can be written as 6n - 1 where n = 1.
Only 2 and 3 cannot be written in one of the two forms: 6n+1 or 6n - 1.

Hence every prime number greater than 3 will be of one of these forms. (mind you, it does NOT mean that every number of the form 6n+1 or 6n-1 will be prime)