The following argument is from Susskind's presentation in The Black Hole War, 2008, p. 152. There is a simple algebraic proof that adding 1 bit to a black hole increases its area by on the order of one Planck unit. (This is omitting factors of order unity; there is actually a factor of 1/4 involved.) Drop in a photon of energy $E$ that encodes one bit. If the photon was well localized, then there might be a lot more than 1 bit carried, since it would enter the black hole at a certain location. So to get just one bit, make $\lambda \sim R$, where $R$ is the Schwazschild radius. (Making $\lambda > R$ doesn't work, because the photon just scatters without being absorbed.) The photon's energy $E=hc/\lambda$ is equivalent to a mass $dm=E/c^2=h/c\lambda$. The Schwarzschild radius grows according to $R=2(G/c^2)m$, causing the area to grow as $4\pi R^2$. Neglecting factors of order unity, the growth in area is $R dR\sim m dm \sim Gh/c^3$, which is the Planck area.
what units is the information measured in
Entropy can be defined as the log of the number of accessible states, $\ln M$, or, in SI units, as $k\ln M$, where $k$ is Boltzmann's constant. If you let $k=1$, then essentially you're measuring entropy in something pretty close to units of bits. (The actual factor is that one bit is $(1/2)\ln 2$ units of entropy if $k=1$.)
