I will answer the question in the title:
How do particles become entangled?
Entanglement is a shorthand way of saying that "one is dealing with a quantum mechanical system which describes in a probabilistic manner the particles' variables, as solutions of specific quantum mechanical equations with specific boundary conditions."
( aside : Entanglement could be attributed to solutions of classical equations: when one has the solution for a planet and its satellite and the boundary conditions are given, if one knows where the satellite is, one also knows where the planet is, both revolving about their barycenter.)
In the framework of atoms and molecules and photons the quantum mechanical equations describe many body systems, and various quantum mechanical models have been developed to deal with quantum mechanical collective phenomena. Quantum mechanical entanglement means that the probability distributions, ( the square of the state function) for measurable behaviors of the particles are completely determined for the system.
The molecules in a crystal itself are entangled because in principle a state function can be written for the crystal.
In interactions, a new solution has to be used. When a proton hits a proton at the LHC the whole interaction, input particles, output particles and all the correlations and angles of the output particles are entangled . The statistical behavior of the interaction is given by the square of the state function describing the interaction and the experiment can measure the probability of production of the Higgs , for example.
If you want to use an example with light, you have again to go to the quantum mechanical level, the quanta of light, the photons. In lasing action, stimulated emission, a collective interaction, the whole process is entangled, producing a coherent light beam that emerges from the multitude of photons in the process.