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If I have a graphic matroid $(E,\mathcal{I})$ for a connected graph $G=(V,E)$.

How would I describe the independent sets of the dual Matroid?

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    You can use the definition i.e., get its basis(spanning trees) and the basis of the dual are it's complements. If you want to do it geometrically, get the dual graph(geometrically dual) first and then compute basis there. – 2017-02-06
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    Thanks for the help! Could you elaborate a little bit further please? – 2017-02-06

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If you can dualize the graph, that's great, but if $G$ is non-planar then that method can't really apply. However, it might be good for your understanding if you take some example planar graphs $G$ and dualize them to get $G^*$, and then look at the independent sets of $G^*$ and see how the corresponding edge sets of $G$ behave.

An independent set is just a set that doesn't have a circuit as a subset. If you have a graphic matroid $M(G)$, can you describe what a circuit of the dual matroid $M^*(G)$ looks like in $G$? If you can describe circuits, you can describe dependent sets too. Then your independent sets are the sets that don't fit your description of dependent sets.