The unusual transient Swift J1644+57 likely resulted from a collimated relativistic jet powered by accretion onto a massive black hole (BH) following the tidal disruption (TD) of a star. Several mysteries cloud the interpretation of this event: (1) extreme flaring and `plateau shape of the X-ray/gamma-ray light curve during the first 10 days after the gamma-ray trigger; (2) unexpected rebrightening of the forward shock radio emission months after trigger; (3) no obvious evidence for jet precession, despite misalignment typically expected between the angular momentum of the accretion disk and BH; (4) recent abrupt shut-off in jet X-ray emission after 1.5 years. Here we show that all of these seemingly disparate mysteries are naturally resolved by one assumption: the presence of strong magnetic flux Phi threading the BH. Initially, Phi is weak relative to high fall-back mass accretion rate, Mdot, and the disk and jets precess about the BH axis = our line of sight. As Mdot drops, Phi becomes dynamically important and leads to a magnetically-arrested disk (MAD). MAD naturally aligns disk and jet axis along the BH spin axis, but only after a violent rearrangement phase (jet wobbling). This explains the erratic light curve at early times and the lack of precession at later times. We use our model for Swift J1644+57 to constrain BH and disrupted star properties, finding that a solar-mass main sequence star disrupted by a relatively low mass, M~10^5-10^6 Msun, BH is consistent with the data, while a WD disruption (though still possible) is disfavored. The magnetic flux required to power Swift J1644+57 is too large to be supplied by the star itself, but it could be collected from a quiescent `fossil accretion disk present in the galactic nucleus prior to the TD. The presence (lack of) of such a fossil disk could be a deciding factor in what TD events are accompanied by powerful jets.[abridged]
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