What happens when a photon hits a beamsplitter?

The crucial word is "beam", in "beam splitter". Beam means an ensemble, in contrast to "photon" which is an individual particle.

A light beam is an ensemble of photons and if it is of a single frequency $\nu$, all photons have energy $E= h*\nu$. A light beam can be split in a beam spliter, i.e. the ensemble of photons can be split into two streams of photons: the intensity of the beam goes down, but the individual photons still have frequency $h*\nu$.

Now one can think of impinging photons one by one on a beam splitter. A photon is described by a wavefunction which when squared will give the probability of finding the photon in a particular (x,y,z). It will go either where one stream went or the other according to the probabilities, but it will be seen as a whole photon of energy $E=h*\nu$.

A single photon is a quantised packet of Electromagnetic energy, the smallest indivisible unit imposed by boundary conditions according to quantum mechanics. In this regime I find it easier to think of the photon as a particle with a 'polarization' degree of freedom which can be horizontal $\left|H\right>$,vertical $\left|V\right>$ or in any linear superposition of the two.

When the photon meets a beam splitter it acts like a quantum particle and takes both paths with some probabilities. Much like the electron taking both paths in Young's famous double slit experiment.

It is not the actual photon being split into two new ones, only the 'particle-like' position wavefunction holds the information about where the photon is. Quantum mechanically, we can treat photons exactly like any other bosonic particle.

Very short and "axiomatic" answer: You indeed can "split" one particle. In QM particles are treated as a "wave functions", maybe it will be more easy for you to imagine a splitting wave. However, only at the point when the photon is detected the particle is measured in one point of space. This is the very foundation of QM and I agree that it's hard to grasp the concept.