A CLEC, competitive local exchange carrier, is described...
A competitive local exchange carrier (CLEC), in the United States and Canada, is a telecommunications provider company (sometimes called a "carrier") competing with other, already established carriers, generally the incumbent local exchange carrier (ILEC).
The source for this quote... is a Wikipedia page.
It is my understanding that the ILEC is a former monopoly and that former monopolies usually exist because they were or are natural monopolies. The introduction of a CLEC does not change the fact that the infrastructure of the ILEC has a high cost to create and that cost is an insurmountable obstacle to creating a parallel infrastructure. In fact, the CLEC typically does not create a parallel infrastructure and instead it will rent use of the original electronic or optical fiber infrastructure of the ILEC.
The basis of competition would seem to originate in infrastructure other than the electronic or optical fiber infrastructure. This might take the form of what used to be a call center, although these people can work from home so physical call centers do not even need to exist. Websites can replace call centers for a business that wants to completely avoid talking to people. What remains is the mental effort to create a website to advertise and support customers of the CLEC and perhaps some retail locations to do essentially the same thing that the website does. CLECs exist on what is a small fraction of the infrastructure of the ILEC. If the infrastructure is small there is not a big domain in which to innovate. If the industry is mature there may not be obvious ways to innovate. If there is no innovation then the large scale operation wins; the ILEC wins.
Obviously the CLEC can exist if the ILEC decides not to force the CLEC out of business through competition. The ILEC wants to be popular with politicians and regulators and politicians want to be popular with voters. The CLEC can thrive if politicians want it to thrive and if the ILEC knows what the politicians want. An different way for politicians to be popular with voters is to regulate the profit of the ILEC and voters will pay lower prices. The ILEC profit might be so low that there will be no CLECs.
Edit: The question is "how does a CLEC innovate?" There were comments that suggest CLECs are not innovating and that CLECs are "innovating better ways of getting your money, not better ways of making the Internet work". These comments seem to take a narrow view of innovation being necessarily connected to technological improvement and at the same time seem to allow that "getting your money" is an (ironic?) innovation. I do not subscribe to the narrow view and the irony. I think the word innovative is appropriate to describe any business that can sustain itself in a competitive environment. My broad view is that every form of differentiation is an innovation whether we can prove the individual differentiating choices are beneficial or not. Since there is uncertainty we cannot rule it out. This question recognizes that the status quo shows that the CLEC business model works, but how and why?