Why/how does an electron emit a photon when decelerating?

This is a hand waving answer.

You ask in the comments: I see that that particle gains/loses energy, but aren't there other ways to do that?

Down in the particle world everything is quantum mechanical and the only things that exist are the standard model particles, which may sometimes act as waves according to strict QM rules.

What are the possible interactions of an electron? Weak and electromagnetic. Weak is orders of magnitude weaker than the electromagnetic ( hence the name) and can be ignored.

Thus any measurable interaction an electron can have has to be electromagnetic. In the microcosm dimension any gaining or losing energy has to go through photons .

This answer is in part an attempt to demonstrate how you often can apply very different perspectives to the same physics question.

In his PhD thesis, Richard Feynman asked this question: "Why does an electron recoil when it emits a photon?" He asked this question from a rather peculiar perspective, in which he assumed that fields do not exist and particles only interact directly."

Your question simply reverses the order of causation: "why does an electron emit a photon when it recoils (decelerates)?"

So, the answer from Feynman's thesis applies nicely to your question. The details of Feynman's answer were incorrect on one point, where Feynman assumed that an electron could not interact with itself. @RonMaimon pointed this out here in Physics.SE, but also why it does not undermine Feynman's main answer.

So what would Feynman's answer be to your question likely would have been this: An electron emits an ordinary photon in response to being struck by a reverse-time ("advanced") photon that has traveled backward in time from some point in the future. "Some point in the future" could range from femtoseconds (or less) away up to billions of years in the future. The real photon that is emitted in response to the recoil-inducing, backwards-in-time photon travels along the classical time path as a normal or "retarded" photon (no I did not make that up), and eventually strikes the very same target that emitted the advanced photon sometime in the future.

The advanced photon solutions, incidentally, were a second set of solutions to Maxwell's equations that traditionally were just ignored as irrelevant, for obvious reasons. That changed when John Archibald Wheeler, Feynman's thesis adviser, suggested them as possible solutions to Feynman's seemingly hopeless quest to create a self-consistent theory of how particles might interact "directly" (whatever that really means), without using any intermediate classical fields.

While I realize this is likely not the type of answer you were expecting, the success of the QED mathematical framework shows that as an answer, it is both self-consistent and correctly predictive. It just approaches the question more form a quantum theory perspective than from a special relativity perspective.

So, to recapitulate: (I always wanted to use that word once [only] in my lifetime!)

Electrons emit ordinary photons when they recoil in response to being struck by an "advanced" photon that has traveled backwards in time. Once emitted, the ordinary or "retarded" photon travels along with the rest of us through ordinary classical time until it completes the circuit -- possibly quickly, possibly very slowly indeed -- by striking the same electron that was the source in the future of the advanced photon.

Addendum 2012-06-13.20:40 EDT - In response to a good comment about causality:

Causality, remarkably, is preserved throughout all of this, as proven in a series of papers by Feynman and Wheeler. And what that says is mostly that our concept of "now" is really quite a bit more complicated than we usually think.

More specifically, quantum particles embedded in ordinary thermodynamic matter require longer integration lengths into what we think of as the past and future -- and those extensions can be very long indeed. I like to think of this fluctuating "toothiness" of quantum particles embedded in the ordinary thermodynamic matter as the "ragged edge of now", but that's just my personal visualization... one that a book author and some Hollywood folks must have had many years before I did. But rows of temporal teeth aside, all I mean by this amusing visualization is that the time distances over which path integrals must be integrated increases as the particles remain quantum for longer periods of time.

Charged particles are permanently coupled to the electromagnetic field, it's an experimental fact and the very essential feature of charges. As any coupled (compound) system, the system (electron + EMF) has its center of mass variables and the "relative motion" (or "internal") variables. Generally, when you act on one of a constituents of a compound system, you transfer energy to its center of mass and to internal variables, both energies being additive. Just like hitting a ball - you push it as a whole and you make it vibrate "internally" (you excite the "shape dynamics", too).

I think when you act on the electron, the previous state of "relative motion" of the compound system (electron + EMF) gets perturbed and one observes relaxation of this perturbation as electromagnetic waves.