What is this *redundant cycle* "thing"? Why is it a matroid? I found the following definition in an introductory exposition about matroids.

Let $G=(V,E)$ be a given (finite) graph. Let $\mathcal{C, D}$ be arbitrary
  collections of cycles in $G$. 
$\mathcal{C}$ is redundant, if every edge in $G$ appears in an even
  number of cycles of $\mathcal{C}$. $\mathcal{D}$ is independent, if
  there is no subset of $\mathcal{D}$ which is redundant.

I am supposed to be showing that the set of independent sets of cycles form a matroid.
I do not know how to proceed here, or what this notion of redundancy is actually referring to. It took me a while to come up with a graph and a redundant cycle set, but I didn't get any intuition of what this is about.
What is the intuition behind these objects? How do I proceed to show that these indeed form a matroid?

Suppose you want to count the cycles in a ladder graph.

Clearly it looks as though a ladder graph with $n$ rungs is made of $n-1$ cycles glued together. However, if you count all possible cycles, you find $\binom n2$ of them (choose any two rungs and take the rectangle between them). The notion of independent cycles formalizes the former intuition; all independent sets of cycles have $n-1$ elements.
