The hot Jupiter WASP-43b has now joined the ranks of transiting hot Jupiters HD 189733b and HD 209458b as an exoplanet with a large array of observational constraints on its atmospheric properties. Because WASP-43b receives a similar stellar flux as HD 209458b but has a rotation rate 4 times faster and a much higher gravity, studying WASP-43b serves as a test of the effect of rotation rate and gravity on the circulation when stellar irradiation is held approximately constant. Here we present 3D atmospheric circulation models of WASP-43b using the SPARC/MITgcm, a coupled radiation and circulation model, exploring the effects of composition, metallicity, and frictional drag. We find that the circulation regime of WASP-43b is not unlike other hot Jupiters, with equatorial superrotation that yields an eastward-shifted hotspot and large day-night temperature variations (~600 K at photospheric pressures). We then compare our model results to observations from Stevenson et al. which utilize HST/WFC3 to collect spectrophotometric phase curve measurements of WASP-43b from 1.12-1.65 microns. Our results show the 5x solar model lightcurve provides a good match to the data, with a phase offset of peak flux and planet/star flux ratio that is similar to observations; however, the model nightside appears to be brighter. Nevertheless, our 5x solar model provides an excellent match to the WFC3 dayside emission spectrum. This is a major success, as the result is a natural outcome of the 3D dynamics with no model tuning, and differs significantly from 1D models that can generally only match observations when appropriately tuned. In sum, these results demonstrate that 3D circulation models can provide important insights in interpreting exoplanet atmospheric observations, even at high spectral resolution, and highlight the potential for future observations with HST, JWST and other next-generation telescopes. ; Comment: 17 pages, 18 figures; accepted to ApJ