Your question is identical to a question posed to Feynman by his father. A two-minute monologue on that is here:
The photons are genuinely created at some moments – and they may be similarly destroyed, too. They're not coming from anywhere and they are going nowhere. The number of photons $N_\gamma$ is simply not conserved in Nature. (Just to be sure, the momentum, angular momentum, and energy are conserved, and this may reduce the number of photons or other particles that may be produced.) And quite generally, whenever a charged particle moves (especially with acceleration) or interacts with others, new photons are almost always created.
In modern physics, we describe this creation of photons by new photon lines attached to the rest of the Feynman diagrams via a Feynman vertex. Classically, we were talking about electromagnetic waves and those were simply created whenever electric charges were accelerating.
Feynman's strategy to explain why it's possible was in terms of an analogy with speaking. A photon is a quantum of light – similarly to the word's being a quantum of speech. If we speak, we can say the word "cat" as many times as we want. We don't have any "word bag" with the words "cat" that could get depleted. We just produce the word – sound waves with the pattern carrying the audio information about the word "cat" – as we speak. In the very same way, sources of electromagnetic radiation keep on producing photons out of nothing. There is no "photon bag" that could ever get depleted.
Similarly, in weak interactions, a W-boson may get created out of nothing – if the aforementioned conservation laws are obeyed (along with the electric charge conservation) and the W-boson may also be destroyed given the same conditions. There are Feynman vertices from which photons or very analogously W-bosons are coming.
Just like particles may be created and annihilated, the list of particle species present in the world may change in more general way. For example, when a neutron decays,
$$ n\to p + e^- +\bar \nu_e,$$
it doesn't prove that the neutron contained the particles from the right hand side. If it did, it wouldn't be a proton but a (much larger) hydrogen atom (plus some strangely attached neutrino). Microscopically, the decay is due to the quark-level decay
$$ d \to u + e^- + \bar\nu_e $$
which changes one fermion into three others. This single "four-fermion interaction" may be further reduced to a Feynman diagram with a virtual W-bosons. The latter diagram has two vertices. In one of them, the down-quark is destroyed and replaced by an up-quark and a negative W-boson. In the other vertex, the W-boson decays to the electron-neutrino pair. This general creation or transformation of particles doesn't violate any valid law of physics.