CUPID-Mo is a bolometric experiment to search for neutrinoless double-beta decay ($0\nu\beta\beta$) of $^{100}$Mo. In this article, we detail the CUPID-Mo detector concept, assembly, installation in the underground laboratory in Modane in 2018, and provide results from the first datasets. The demonstrator consists of an array of 20 scintillating bolometers comprised of $^{100}$Mo-enriched 0.2 kg Li$_2$MoO$_4$ crystals. The detectors are complemented by 20 thin cryogenic Ge bolometers acting as light detectors to distinguish $\alpha$ from $\gamma$/$\beta$ events by the detection of both heat and scintillation light signals. We observe good detector uniformity, facilitating the operation of a large detector array as well as excellent energy resolution of 5.3 keV (6.5 keV) FWHM at 2615 keV, in calibration (physics) data. Based on the observed energy resolutions and light yields a separation of $\alpha$ particles at much better than 99.9\% with equally high acceptance for $\gamma$/$\beta$ events is expected for events in the region of interest for $^{100}$Mo $0\nu\beta\beta$. We present limits on the crystals' radiopurity ($\leq$3 $\mu$Bq/kg of $^{226}$Ra and $\leq$2 $\mu$Bq/kg of $^{232}$Th). Based on these initial results we also discuss a sensitivity study for the science reach of the CUPID-Mo experiment, in particular, the ability to set the most stringent half-life limit on the $^{100}$Mo $0\nu\beta\beta$ decay after half a year of livetime. The achieved results show that CUPID-Mo is a successful demonstrator of the technology - developed in the framework of the LUMINEU project - selected for the CUPID experiment, a proposed follow-up of CUORE, the currently running first tonne-scale cryogenic $0\nu\beta\beta$ experiment.