Saturation current in photoelectric effect
In the photoelectric effect, photons incident on the cathode cause the emission of electrons. Assuming there is a sufficient electric field, these electrons will make their way across to the anode, contributing current.
For simplicity, let's assume every photon generates a photo-electron. Then if $N$ photons per second hit the cathode, the current will be carried by a total of $N$ electrons per second. We always assume there are "infinitely many" electrons waiting for their turn, and the thing limiting the current is how many electrons get "released" from the cathode (i.e. how many photons hit the cathode).
Current is charge per unit time. If the electron has a charge $q_e$, then $N$ electrons per second carry a current
$$I = N\; q_e$$
There is nothing here about the velocity of the electrons... not about the time it takes them to cross the gap. If they went 100 times faster, it would not change the number of electrons crossing the gap per second. That number is determined by "how many start the trip per second" and "how many don't make it". The second of these explains that the curve starts out not completely flat: very slow electrons may not make it, especially with a small retarding potential. But once they go fast enough to fully escape, their final speed really doesn't matter. And neither does the transit time in the wire.
Did you ever calculate how slowly electrons move in a current carrying wire (for example, copper wire)? While the electrical signal is very fast, the drift velocity of the electrons themselves is very very slow... because there are so many electrons per unit volume. But that is only tangentially relevant here.
I want to add to the answer by Floris, which explains the phenomenon correctly. Ask yourself the meanings of stopping potential and the saturation current.
The stopping potential is determined by the energy of the photons minus the work function of the material in question. Let's idealize the situation: For example, the work function of copper is 4.7 eV. Say if you have a light source and shine the copper with light of energy 5.7 eV then the stopping potential will be roughly -1 V. That is because, a photon with energy 5.7 eV hits an electron, electron uses 4.7 eV to free itself from the copper metal and gains 1 eV as kinetic energy. But it can never make it to the other electrode because it is stopped by the voltage. Increase the potential (that is go towards 0 and positive values) then the electron can make it to the other electrode and you get current.
In reality, however some of the electrons looses their kinetic energy even before they can escape from copper. That is, they end up with kinetic energies less than 1 eV and therefore they need a "push" to be able to reach the other electrode. As you increase the voltage you give that push to these poor electrons and make them reach the electrode and hence increase the current.
However at some point the electrons which can make to the other electrode will be saturated because the light source can produce certain amount of photons and therefore certain amount of photoelectrons. The scientific term for "amount of photons" is the intensity of the light source. If you have one more equivalent light source and lit it then you will get the saturation current roughly doubled.
There nothing much to add to the beautifully scripted answers above.
Basically the number of photo electrons emitted depends on the intensity of incident radiation, while the maximum kinetic energy that an emitted photo electron can posses is a function of the potential across the collector plates. Besides, it is obvious that the maximum number of photo electrons which can be reach the collector plate depends on the intensity, therefore for a given intensity (i.e there is a given maximum number of photo electrons which are emitted), whether these reach the collector (overcoming repulsions by space charge) is depends on the potential.
Therefore the number of photoelectrons reaching the collector plate reaches a unique constant value (equaling the maximum number of photoelectrons based on the intensity of incident radiation).