Dissipative Kerr cavity solitons (CSs) are localized temporal structures generated in coherently-driven Kerr resonators that have attracted widespread attention for their rich nonlinear dynamics and key role in the generation of optical frequency combs. Akin to the complexity of these dissipative solitons in mode-locked lasers, the nonlinear dynamics of the CSs present distinctive evolutionary behaviors that may create new potential for understanding interdisciplinary nonlinear problems. Here, we leverage real-time spectroscopy to study the transient behaviors of the CSs in a Kerr fiber cavity with coherent driving. The real-time spectroscopy is implemented with the emerging dispersive Fourier transform (DFT) technology with a large dispersion of -10.2 ns/nm, which provides a sampling spectral resolution of ~1 pm. Under perturbations, the complete birth-to-annihilation process of the CS is visualized in real time as the Kerr fiber cavity specifically locked around the boundary of modulation instability (MI) and bistable regimes. Fruitful transient dynamics are observed, including MI, soliton breathing, stationary CS and its annihilation. The mechanism of the observed transient dynamics is theoretically studied through numerical simulation, finding that the cavity detuning variation resulted from the external perturbation plays a dominant role in the evolution of the CS. More importantly, there exists a visible energy drop accompanied with the CS breathing, wherein the collision between the solitons triggers the subsequent drift and annihilation of the CS. The spectral interferograms of multiple CSs that are analyzed by their field autocorrelation also verify the annihilation of the CS.