A matroid gives a general description of those problems where a greedy algorithm provides an optimal solution. Intuitively, they state that you can build a solution step-by-step (this is given by two of the matroid properties, "the empty set is contained in the matroid" and "if a set is contained in the matroid, every subset is contained as well"), and if the solution can be further optimized, it doesn't matter which path you take to the optimum (this is the exchange property of matroids).
The example is, in my opinion, not really illustrative. What they mean is: Say you are trying to find a basis for the vector space generated by the rows of a matrix. The basis is a finite subset of these rows, and a basis is a maximal linear independent set. Now, we know that the empty set is (trivially) linearly indepedent, and every subset of a linearly independent set is also linearly indepedent. Also, if a set is linear indepedenent but not a basis, it doesn't matter which exact new vector we add (as long as it's linearly independent) - every valid extension will lead us to a basis.