With competition, you must advertise your strengths and downplay your weaknesses. The reason is so that you can maintain an edge over your teammates. The team functions like a poker competition. The competition resolves which member has the strongest set of skills so that the most difficult work can be assigned to the one with the greatest skill. This minimizes the risk of failure.
With collaboration, you must advertise your weaknesses and downplay your strengths. The reason is so that the difficult high-risk work can be assigned or divided efficiently among the group. It ensures the difficult work is assigned away from those with the least skill, which is another way to minimize the risk of failure. It's also a demonstration of honesty for the purpose of establishing trust. Trust is critical for fostering an environment for collaboration.
I'm not sure I fully agree with this breakdown, but it is a neat way to look at the issue.
The key problem is that you want tasks to be assigned to people to match skills. But skill is an unknown and must be "discovered". Collaboration sounds better, but what if competition is actually more accurate at skill discovery?
The big problem comes when their is a signalling mismatch. If someone's ability to signal strength outdoes their actual strength in competition, or someone's ability to signal weakness overstates their weakness.
The actual comment comes up in a discussion of gender approaches to teamwork (guess which gender is assumed to use which model!), but I think the gender angle is unnecessary. I think the two models are worth looking at on their own.