In this paper we study a key phase in the formation of massive galaxies: the transition of star forming galaxies into massive (M_stars~10^11 Msun), compact (r_e~1 kpc) quiescent galaxies, which takes place from z~3 to z~1.5. We use HST grism redshifts and extensive photometry in all five 3D-HST/CANDELS fields, more than doubling the area used previously for such studies, and combine these data with Keck MOSFIRE and NIRSPEC spectroscopy. We first confirm that a population of massive, compact, star forming galaxies exists at z~2, using K-band spectroscopy of 25 of these objects at 2.0<z<2.5. They have a median NII/Halpha ratio of 0.6, are highly obscured with SFR(tot)/SFR(Halpha)~10, and have a large range of observed line widths. We infer from the kinematics and spatial distribution of Halpha that the galaxies have rotating disks of ionized gas that are a factor of ~2 more extended than the stellar distribution. By combining measurements of individual galaxies, we find that the kinematics are consistent with a nearly Keplerian fall-off from V_rot~500 km/s at 1 kpc to V_rot~250 km/s at 7 kpc, and that the total mass out to this radius is dominated by the dense stellar component. Next, we study the size and mass evolution of the progenitors of compact massive galaxies. Even though individual galaxies may have had complex histories with periods of compaction and mergers, we show that the population of progenitors likely followed a simple inside-out growth track in the size-mass plane of d(log r_e) ~ 0.3 d(log M_stars). This mode of growth gradually increases the stellar mass within a fixed physical radius, and galaxies quench when they reach a stellar density or velocity dispersion threshold. As shown in other studies, the mode of growth changes after quenching, as dry mergers take the galaxies on a relatively steep track in the size-mass plane.
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