The most common way to discover extragalactic fast transients, which fade within a few nights in the optical, is via follow-up of gamma-ray burst and gravitational-wave triggers. However, wide-field surveys have the potential to identify rapidly fading transients independently of such external triggers. The volumetric survey speed of the Zwicky Transient Facility (ZTF) makes it sensitive to objects as faint and fast fading as kilonovae, the optical counterparts to binary neutron star mergers, out to almost 200 Mpc. We introduce an open-source software infrastructure, the ZTF REaltime Search and Triggering, ZTFReST, designed to identify kilonovae and fast transients in ZTF data. Using the ZTF alert stream combined with forced point-spread-function photometry, we have implemented automated candidate ranking based on their photometric evolution and fitting to kilonova models. Automated triggering, with a human in the loop for monitoring, of follow-up systems has also been implemented. In 13 months of science validation, we found several extragalactic fast transients independently of any external trigger, including two supernovae with post-shock cooling emission, two known afterglows with an associated gamma-ray burst (ZTF20abbiixp, ZTF20abwysqy), two known afterglows without any known gamma-ray counterpart (ZTF20aajnksq, ZTF21aaeyldq), and three new fast-declining sources (ZTF20abtxwfx, ZTF20acozryr, ZTF21aagwbjr) that are likely associated with GRB200817A, GRB201103B, and GRB210204A. However, we have not found any objects that appear to be kilonovae. We constrain the rate of GW170817-like kilonovae to R < 900 Gpc(-3) yr(-1) (95% confidence). A framework such as ZTFReST could become a prime tool for kilonova and fast-transient discovery with the Vera Rubin Observatory.