How do particles become entangled? A person asked me this and I'm just a lowly physical chemist.
I used a classical analogy. (How good or bad is this and how to fix it?)
Basically, light has a net angular momentum of zero, insofar as it is not polarized into its left and right plane polarized forms until it hits a crystalline structure.
However, once it does hit such a structure, we have left and right plane polarized light--that is left and right photon beams.
Since the original light was not polarized, this polarization (left and right) is inherent in the light.  The original light is a superposition of left and right polarized light, each with a total angular momentum of -1 and 1, so that they result in the total zero polarized incident light.
Thus, they are entangled to lower their spin (quantum angular momentum) to zero.  Once we measure one of the particles in the superposition, we know the other by conservation of angular momentum.
Is this close?
 A: Other answers paraphrase it well in technical terms. 
It might be easier to see if you remember that when two particles interact they must do so in a way so that the momentum, energy, spin, etc. are conserved. 
After the interaction the two particles still remain in a superposition state but if you measure one of them after an interaction you can find out what state the other particle must be in order to adhere to conservation laws.
So this means whenever to particles interact some form of entanglement must happen.
A: 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.
A: What you've described seems to be one example. In general, interactions between two quantum systems will put the system into some joint state which will generically be entangled. For example, if you have two spins coupled with a spin-spin coupling, then the ground state of that (total) two spin system has some entanglement.
EDIT: BTW, your analogy is not classical but actually quantum. And it's not just an analogy but captures the physics of entanglement accurately. People think of light polarization as classical since they learn it in an electrodynamics course, but essentially it is quantum behaviour -- for "massless" stuff like photons, the classical description has some inherently quantum properties :-)
