We study the spatially resolved star formation history (SFH) and molecular gas distribution of 58 nearby galaxies, using integral field spectroscopy from the CALIFA survey and CO J=1-0 intensity mapping from the CARMA EDGE survey. We use the 4000{AA} break (D_n_(4000)), the equivalent width (EW) of the H{delta} absorption line (EW(H{delta}_A_)), and the EW of the H{alpha} emission line (EW(H{alpha})) to measure the recent SFH of these galaxies. We measure radial profiles of the three SFH indicators and molecular gas mass surface density, from which we measure the level of centrally enhanced star formation and the molecular gas concentration. When we separate our galaxies into categories of barred (17 galaxies), unbarred (24 galaxies), and merging/paired (17 galaxies), we find that the galaxies that have centrally enhanced star formation (19/58) are either barred (13/19) or in mergers/pairs (6/19) with relatively high molecular gas concentrations. A comparison between our barred galaxies and a snapshot of a hydrodynamic N-body simulation of a barred galaxy shows that the current theory of bar formation and evolution can qualitatively reproduce the main features of the observed galaxies in our sample, including both the sharp decrease of stellar age in the galactic centre and the gradual decrease of age with increasing distance from centre. These findings provide substantial evidence for a picture in which cold gas is transported inwards by a bar or tidal interaction, which leads to the growth and rejuvenation of star formation in the central region.