What is the significance of ear decompositions for non-graphic matroids? On Wikipedia there is subsection in the article on ear decompositions of graphs titled "Matroids":

Now as defined above, the circuits of a matroid can not always be listed to satisfy the above two lemmas. Arranging the set of all circuits $\mathcal{C}$ for any matroid $M$ to satisfy the first lemma is equivalent to finding a path in the intersection graph $I(\mathcal{C})$ of $\mathcal{C}$ that traverses every vertex of $I(\mathcal{C})$ (this is the graph with vertex set $\mathcal{C}$ and adjacencies iff two circuits have non-empty intersection) therefore this constrains us to dealing with matroids that are a direct sum of a connected matroid and any free matroid as answered here (WLOG it suffices to require $M$ be connected since any free summands contribute no circuits and can afterword be added back without changing the ear decompositions).
Though what about the second lemma? Which matroids satisfy that property? I'm also not entirely sure what they mean by contract the circuits, do they mean; given an indexing $C_1,\ldots C_n$ of $\mathcal{C}$ such that: $\forall 0<i\leq n\exists j<i:C_i\cap C_j\neq\emptyset$, that for every integer $1\leq i\leq n$ the circuit $C_i$ of $M$ must be a circuit of the matroid $M'=M/C_1/C_2\ldots /C_{i-1}=M/(C_1\cup C_2\cup \cdots\cup C_{i-1})$ s.t. the contraction of $\small M=(U,\mathcal{I})$ by $\small X\subseteq U$ is defined $\small M/X=(U\setminus X,\{S\subseteq U\setminus X:S\cup X\in\mathcal{I}\})$? Further if this is what is meant, then intuitively what do these ear decompositions "mean" or what do "they correspond do"? Perhaps viewing the matroid $M$ as an abstract simplicial complex so independent sets in $\mathcal{I}$ are the faces of the complex and $U$ is the complexes' vertex set might assign some intuition to these definitions?
So in short im asking - what is the significance of defining 'ear decompositions' for matroids (i.e. does this lead to some geometric intuition using complexes as I mentioned? or perhaps does it allow one to better study the circuits? why ever bother defining these 'ear decompositions'?) also what matroids even have these decompositions (i.e. what connected matroids have circuits that can be arranged to satisfy the two lemmas mentioned in the Wikipedia article)?
 A: A matroid has an ear decomposition if and only it is connected (this answers your last question).  This is a generalization of the fact that a graph has an ear-decomposition if and only if it is $2$-connected.  Note that graph $2$-connectivity corresponds to matroid connectivity in the sense that $M(G)$ is connected if and only if $G$ is $2$-connected, where $M(G)$ is the cycle matroid of $G$.
As for your other questions, ear-decompositions can be used in inductive arguments to prove things about connected matroids or $2$-connected graphs (remove an ear and apply induction).  They can also be thought of as a way to build all $2$-connected graphs or all connected matroids. In short, they are another way to think about connected matroids or $2$-connected graphs.
It is easy to see that the number of ears in every ear-decomposition of a graph $G$ is the same and is equal to $|E(G)|-|V(G)|+1$ (deleting one edge from each ear yields a spanning tree).  If you view $G$ as a simplicial complex, this is the dimension of the homology group of $G$. So if you like, you can view the number of ears as a geometric invariant.
Lastly, other graph properties can be expressed as having an ear decomposition satisfying some extra conditions.  For example, the following is a classic theorem of Lovász.
Theorem (Lovász).
A $2$-connected graph is factor-critical if and only if it has an ear-decomposition in which all its ears have an odd number of edges.
Here a graph $G$ is factor-critical if $G-v$ has a perfect matching for every $v \in V(G)$.  This notion has been extended to binary matroids by Yohann Benchetrit and András Sebő.  See this
post by Yohann Benchetrit on the Matroid Union Blog for more information.
