You know how strobe light photography lets you 'freeze' very fast moving things so you can study what they look like mid motion. This is used for basically that, except for much much smaller things moving much much faster.
The wavelength at the size of a watermolecule in the range of Exahertz x-ray rather implies very precise laser pulses because the focal point is proportional to the wavelength. It is also relevant for energy transfer into molecules at resonant frequencies.
from the last paragraph linked above:
> For example, attosecond pulses can be used to push molecules, which emit a measurable signal. The signal from the molecules has a special structure, a type of fingerprint that reveals what molecule it is, and the possible applications of this include medical diagnostics.
Basically it's a more precise higher energy X-Ray laser.
I believe the fast turning on and off is a byproduct of a basic method (high-harmonic generation). They do stress the importance of short pulses, but this again may have to do with decoherence of the focal point and not so much the speed of electrons inside the molecule, which is only a model (i.e. relativistic) and remains to be investigated with this new method.
Another benefit of pulsing light really fast is that you can more easily perform studies of really delicate things (eg proteins) because you effectively have higher control over the amount of energy you're pouring into the sample. That isn't directly what they were working on here, but a similar extremely high frequency pulsing is one of the things that makes free-electron lasers 'next generation' compared to synchrotrons.
To add to what was said in the other answers, turning light on and off very quickly broadens the source so that every frequency is present (i.e. take the Fourier transform of a delta) but still focasable to a narrow point.
