We present first results from a series of $NuSTAR$ observations of the black hole X-ray binary V404 Cyg obtained during its summer 2015 outburst, primarily focusing on observations during the height of this outburst activity. The $NuSTAR$ data show extreme variability in both the flux and spectral properties of the source. This is partly driven by strong and variable line-of-sight absorption, similar to previous outbursts. The latter stages of this observation are dominated by strong flares, reaching luminosities close to Eddington. During these flares, the central source appears to be relatively unobscured and the data show clear evidence for a strong contribution from relativistic reflection, providing a means to probe the geometry of the innermost accretion flow. Based on the flare properties, analogies with other Galactic black hole binaries, and also the simultaneous onset of radio activity, we argue that this intense X-ray flaring is related to transient jet activity during which the ejected plasma is the primary source of illumination for the accretion disk. If this is the case, then our reflection modeling implies that these jets are launched in close proximity to the black hole (as close as a few gravitational radii), consistent with expectations for jet launching models that tap either the spin of the central black hole, or the very innermost accretion disk. Our analysis also allows us to place the first constraints on the black hole spin for this source, which we find to be $a\ast >0.92$ (99% statistical uncertainty, based on an idealized lamp-post geometry).