Evidence is mounting that recent multi-wavelength detections of fast blue optical transients (FBOTs) in star-forming galaxies comprise a new class of transients, whose origin is yet to be understood. We show that hydrogen-rich collapsing stars that launch relativistic jets near the central engine can naturally explain the entire set of FBOT observables. The jet-star interaction forms a mildly-relativistic shocked jet (inner cocoon) component, which powers cooling emission that dominates the high velocity optical signal during the first few weeks, with a typical energy of ∼ 10 50 − 10 51 erg. During this time, the cocoon radial energy distribution implies that the optical lightcurve exhibits a fast decay of L ∝ ∼ t −2.4 . After a few weeks, when the velocity of the emitting shell is ∼ 0.01 c, the cocoon becomes transparent, and the cooling stellar envelope governs the emission. During this transition, discontinuities may emerge in the lightcurve and spectra, as have been observed in a few FBOTs, such as AT2018cow. The interaction between the cocoon forward shock and the dense circumstellar winds generates synchrotron selfabsorbed emission in the radio bands, featuring a steady rise on a month timescale. After a few months, if the jet successfully pierces the stellar envelope, it decelerates, enters the observer's line of sight, and powers the peak of the radio lightcurve, which rapidly decays thereafter. The jet (and the inner cocoon) become optically thin to X-rays ∼ day after the collapse, allowing X-ray photons to diffuse from the central engine that launched the jet to the observer. Cocoon cooling emission is expected at higher volumetric rates than gamma-ray bursts (GRBs) by a factor of a few, similar to FBOTs. We rule out uncollimated outflows, however both GRB jets and failed collimated jets are compatible with all observables.