What is the purpose of a 1 Ω resistor to ground in the output filter of a voltage regulator?
The first easy to use 'one component' regulators to be produced, like the 7805, were NPN follower based. This meant they had a high(ish) dropout voltage. The low output impedance, and the unavailability of low ESR capacitors at the time, meant that they were stable into any pretty much any output capacitor.

To meet the demand for lower dropout voltage, PNP output stages were then used. These had a higher output impedance. When used with high ESR aluminium electrolytics these were usually stable for a range of values. The ESR of the capacitor allowed some non-phase-shifted (or fast) feedback to the control circuitry, which allowed them to remain stable.

Unfortunately, shortly after the introduction of LDOs, ceramic capacitors started getting big enough to be usable after regulators, at the same time as miniaturisation was demanding them. Designers started to use LDOs with very low ESR caps. This removed the fast feedback, and some early LDO designs became unstable as a result. 

The series resistor in your diagram suggests that LP2951 is one of these older LDO designs, that requires a minimum ESR on the output capacitor to be stable. It's not needed for an electrolytic output cap, but will be needed for ceramic.

Newer LDOs have been designed to be stable with low ESR caps. These can be identified by explicit claims on the data sheet that they are stable with ceramic capacitors.