We report the discovery of six active galactic nuclei (AGNs) caught "turning on" during the first nine months of the Zwicky Transient Facility (ZTF) survey. The host galaxies were classified as low-ionization nuclear emission-line region galaxies (LINERs) by weak narrow forbidden line emission in their archival SDSS spectra, and detected by ZTF as nuclear transients. In five of the cases, we found via follow-up spectroscopy that they had transformed into broad-line AGNs, reminiscent of the changing-look LINER iPTF16bco. In one case, ZTF18aajupnt/AT2018dyk, follow-up Hubble Space Telescope ultraviolet and ground-based optical spectra revealed the transformation into a narrow-line Seyfert 1 with strong [Fe vii, x, xiv] and He ii λ 4686 coronal lines. Swift monitoring observations of this source reveal bright UV emission that tracks the optical flare, accompanied by a luminous soft X-ray flare that peaks ∼60 days later. Spitzer follow-up observations also detect a luminous mid-infrared flare, implying a large covering fraction of dust. Archival light curves of the entire sample from CRTS, ATLAS, and ASAS-SN constrain the onset of the optical nuclear flaring from a prolonged quiescent state. Here we present the systematic selection and follow-up of this new class of changing-look LINERs, compare their properties to previously reported changing-look Seyfert galaxies, and conclude that they are a unique class of transients well-suited to test the uncertain physical processes associated with the LINER accretion state.