First, he pointed out that the integration of social functionality is redefining what it means to be a "successful" site. From a social psychology view, communities are dynamic entities, continually evolving as the membership waxes and wanes and the focus of the core membership changes. As such, a site built to support a community element must evolve in parallel with its community. While previously success meant that a site had been well constructed and implemented, now it means that the site is able to continually learn from the evolving community and adapt to support the community's needs. Success is no longer something which is attained but a goal to be continually sought. It also means that a site can really only be considered done when it is being taken down. If the site remains up, it must be continually refactored, expanded, and improved as though it were a living thing itself.
For site design, the implications of this idea are that the site must be architected in such a way that adding or changing features of the site is not prohibitively complex and that the site incorporate monitoring and feedback mechanisms by which the site managers can evaluate the changing needs of the community and receive direct input from the community members as to what they need. In essence, the site must provide a means of determining what needs to change and ensure that those changes can be implemented without necessitating a complete tear-down of the site.
The second point Clay made was that users of a community site are going to be distributed across a range of involvement, from the very highly involved individuals which drive a community to the occasionally involved users who may make one or two posts/commits/etc over the life of the community. Further, the vast majority of contributions will come from a small core of individuals and the bulk of individuals will make only a small number of contributions, as conceptualized by the Long Tail concept or the Pareto aka "80/20" Distribution. Clay was speaking specifically of open-source development projects like the Drupal community, but I believe the same distributions will hold true whether involvement is measured in commits to a code repository or discussion posts.
In this aspect, the goals, then, are to provide tools which enable these high-committing individuals and to simultaneously reduce the barriers to contribution which could daunt the long-tail contributors. For the former group, this could mean opening access to system APIs or providing special "power-user" accounts with more access than is granted to the common user. For the latter group, this could mean enabling anonymous commenting so that users can be involved in the community without registering or ensuring that functionality is clear and accessible with minimal clicks so that users don't get lost or loose interest en route to contributing.
In essence, we must do what we've always known we needed to do: know and understand our users and grow along with them. If we don't, they won't be our users for long.