How can aerogel be lighter than air?

While the summary you cited is a convenient and easy to understand phrase, it is a paraphrase of another cited paper:

• Sun H., Xu Z., Gao C., "Multifunctional, Ultra-Flyweight, Synergistically Assembled Carbon Aerogels", Adv. Mater. 25 (2013) 2554–2560.

The paper says:

The density was calculated by the weight of solid content without including the weight of entrapped air divided by the volume of aerogel (the density measured in a vacuum is identical to that in the air)

So indeed the other answers are correct: the air is not factored into the density, presumably so aerogels can be compared objectively (despite those at higher altitudes and lower humidity being measured less dense).

They're not lighter than air (or almost certainly are not). The mass of $1\,\mathrm{m^3}$ of such a material, in air, is $\rho_m f_m + \rho_a (1 - f_m)$, where $\rho_m$ is the density of the structure, $\rho_a$ is the density of air, and $f_m$ is the proportion of the bulk volume of the material which is structure rather than air. I think that $f_m \approx 0.002$ for a typical aerogel. The density they are quoting is then $\rho_m f_m$.

If one measured the density of a huge led zeppelin with a vacuum inside (like is done with aerogels; the until now lightest aerogel is aerographene, with a density of about 13% of air) this density could be lower than air if the led zeppelin is big enough. Can you imagine the led zeppelin floating when it's filled with air?