While optical surveys regularly discover slow transients like supernovae on their own, the most common way to discover extragalactic fast transients, fading away in a few nights, is via follow-up observations of gamma-ray burst and gravitational-wave triggers. However, wide-field surveys have the potential to also identify rapidly fading transients independently of such external triggers. The volumetric survey speed of the Zwicky Transient Facility (ZTF) makes it sensitive to faint and fast-fading objects as kilonovae, the optical counterparts to binary neutron stars and neutron star-black hole mergers, out to almost 200Mpc. We introduce an open-source software infrastructure, the ZTF REaltime Search and Triggering, ZTFReST, designed to identify kilonovae and fast optical transients in ZTF data. Using the ZTF alert stream combined with forced photometry, we have implemented automated candidate ranking based on their photometric evolution and fitting to kilonova models. Automated triggering of follow-up systems, such as Las Cumbres Observatory, has also been implemented. In 13 months of science validation, we found several extragalactic fast transients independent of any external trigger (though some counterparts were identified later), including at least one supernova with post-shock cooling emission, two known afterglows with an associated gamma-ray burst, two known afterglows without any known gamma-ray counterpart, and three new fast-declining sources (ZTF20abtxwfx, ZTF20acozryr, and ZTF21aagwbjr) that are likely associated with GRB200817A, GRB201103B, and GRB210204A. However, we have not found any objects which appear to be kilonovae; therefore, we constrain the rate of GW170817-like kilonovae to $R < 900$Gpc$^{-3}$yr$^{-1}$. A framework such as ZTFReST could become a prime tool for kilonova and fast transient discovery with the Vera C. Rubin Observatory.