This is of course just a hyperbole, especially in its original context.
For one thing, people routinely use representation theory to study groups of matrices. That is, represent the group of matrices by other matrices, acting on other spaces. If matrices were this elementary "thoroughly understood" object, one presumably wouldn't have needed to do that.
The real power of the representation theory is not that its matrices, but the notion of irreducibility. This allows you to decompose a complex action into simpler blocks and understand it that way.
For one thing, people routinely use representation theory to study groups of matrices. That is, represent the group of matrices by other matrices, acting on other spaces. If matrices were this elementary "thoroughly understood" object, one presumably wouldn't have needed to do that.
The real power of the representation theory is not that its matrices, but the notion of irreducibility. This allows you to decompose a complex action into simpler blocks and understand it that way.