The photon theory that explains the photoelectric effect proposes that it is the interaction of individual photons with individual electrons; that is, each interaction involves one photon and one electron. Einstein's Photoelectric Equation
hf=ϕ+KEMAX describes this; it is a way of looking at the interaction between one photon and one electron in terms of the conservation of energy. The incoming photon has energy
hf. Some of this is needed to overcome the work function
ϕ of the metal; the rest is transferred to the kinetic energy of the electron.
So, onto your actual question... Increased intensity just meand more photons (since they all have the same frequency). The above equation shows that the maximum kinetic energy of any emitted electron is
hf−ϕ; there may be more of them if you increase the intesity (or fewer of them if you decrease the intesity), but the maximum kinetic energy of each emitted electron only depends on the frequency of the light and the work function of the metal.
However, the fact that increased intensity means more photons (again, assuming the frequency of the light is unchanged) means that increased intensity leads to more emitted electrons, and therefore a larger current in a photocell.
Do you appreciate why the above equation includes a term for the maximum kinetic energy of an emitted electron, rather than simply the kinetic energy of such an electron?