We present spectroscopy and photometry of the LMC supersoft binary system RX J0513.9-6951. We derive a refined spectroscopic period of P = 0.761 +/- 0.004 day, which is consistent with the value obtained from long-term photometric monitoring (P = 0.76278 +/- 0.00005 day). We see bipolar outflow components of He II and H beta, with velocities of similar to 3800 km s(-1), strongly suggesting that the compact object is a white dwarf. Using all the available optical and X-ray data, we construct a theoretical model to explain the principal features of the unusual variability of this source. In particular, we note that X-ray outbursts have only been seen at times of optical minima. From this, we conclude that the most likely cause of the X-ray outbursts is a photospheric contraction during a nuclear shell burning phase, rather than a thermonuclear flash or shocked emission. The system probably comprises a relatively massive white dwarf accreting at a high rate (similar to 10(-6) M(.) yr(-1)) from an evolved donor star and is observed close to pole-on.