One of the issues with COVID-19 is the fact that it is transmissible even by people who are asymptomatic. You've probably heard about R0, which is the basic reproduction rate of a disease, that takes into account transmission modes and contact rate, among other things. I'm not an epidemiologist, but I'm operating under the assumption that someone who is obviously sick will infect relatively fewer people than someone who is asymptomatic, assuming similar contagiousness - the sick person should quarantine themself and clearly take measures to avoid spreading their sickness. A seemingly well individual, on the other hand, may not reduce their contact rate and unwittingly spread the disease.
Under this assumption, consider what happens when the incubation time is equal to the doubling rate. Suppose we have 1000 individuals who have the disease, but don't know it yet because they are still in the incubation period. 5.8 days later, there are now 2000 people with the disease, 1000 of which now feel sick, and 1000 of which still feel fine. Those 1000 sick people quarantine themselves, dramatically lowering their rate of infecting others, let's say to 0 for the sake of simplicity. 5.8 days later, the 1000 asymptomatic people now feel sick, after having infected another 1000 people, although the 1000 already-sick individuals haven't infected anyone else. At this rate, we have a linear increase in the number of cases, since old cases are quarantining themselves at the same rate new cases are spread - the number of people spreading the disease unwittingly remains constant!
I've made several assumptions here, like that the transmission rate for asymptomatic individuals is higher than that of symptomatic individuals (although some symptoms, like cough, increase infectivity, and not all symptomatic individuals quarantine appropriately). I'm also not sure this is really an appropriate use of "doubling time" - strictly speaking, if 1000 people quarantine, and 1000 asymptomatic people infect 1000 others, it is not a doubling of cases. The statement also seems a bit of tautological red herring - if you are able to lengthen the doubling time, you have already flattened the curve, regardless of the incubation period. But I think the general idea is that a longer doubling time can lessen the impact of asympomatic, but contagious individuals spreading the disease.