In this paper, we study the onset process of a solar eruption on 2015 February 21, focusing on its unambiguous precursor phase. With multiwavelength imaging observations from the Atmospheric Imaging Assembly (AIA), definitive tether-cutting (TC) reconnection signatures, i.e., flux convergence and cancellation, bidirectional jets, and topology change of hot loops, were clearly observed below the pre-eruption filament. As TC reconnection progressed between the sheared arcades that enveloped the filament, a channel-like magnetic flux rope (MFR) arose in multiwavelength AIA passbands wrapping around the main axis of the filament. With the subsequent ascent of the newborn MFR, the filament surprisingly split into three branches. After a 7 hr slow-rise phase, the high-lying branch containing the MFR abruptly accelerated causing a two-ribbon flare; while the two low-lying branches remained stable forming a partial eruption. Complemented by kinematic analysis and decay index calculation, we conclude that TC reconnection played a key role in building up the eruptive MFR and triggering its slow rise. The onset of the torus instability may have led the high-lying branch into the standard eruption scenario in the fashion of a catastrophe.