what does the redundancy of a code means?

Question

I was reading a paper about a transposition and single deletion error correcting code and they claim that the redundancy of the code was only $\log(6n-3)$ bits.

But what does that means? I was trying to get that from the proof of that fact but what they proved instead was that

there exists a such (with the structure they propose) code whose redundancy is at most $\log(6n-3)$ bits

and in the proof itself they just said that the cardinality of the code is greater or equal than $\frac{2^n}{6n-3}$.

How does that proved the hypothesis?

Also, if I have my own code how do I compute the redundancy (or a reasonable bound on it?)