We present ultra-violet (UV) to mid-infrared (MIR) observations of the long-lasting Type IIn supernova (SN) 2013L obtained by the Carnegie Supernova Project II beginning two days after discovery and extending until +887 days (d). The SN reached a peak r-band absolute magnitude of ≈-19 mag and an even brighter UV peak, and its light curve evolution resembles that of SN 1988Z. The spectra of SN 2013L are dominated by hydrogen emission features, characterized by three components attributed to different emission regions. A unique feature of this Type IIn SN is that, apart from the first epochs, the blue shifted line profile is dominated by the macroscopic velocity of the expanding shock wave of the SN. We are therefore able to trace the evolution of the shock velocity in the dense and partially opaque circumstellar medium (CSM), from ∼4800 km s-1 at +48 d, decreasing as t-0.23 to ∼2700 km s-1 after a year. We performed spectral modeling of both the broad- and intermediate-velocity components of the Hα line profile. The high-velocity component is consistent with emission from a radially thin, spherical shell located behind the expanding shock with emission wings broadened by electron scattering. We propose that the intermediate component originates from preionized gas from the unshocked dense CSM with the same velocity as the narrow component, ∼100 km s-1, but also that it is broadened by electron scattering. These features provide direct information about the shock structure, which is consistent with model calculations. The spectra exhibit broad O » I and [O » I] lines that emerge at ≳ +144 d and broad Ca » II features. The spectral continua and the spectral energy distributions (SEDs) of SN 2013L after +132 d are well reproduced by a two-component black-body (BB) model; one component represents emitting material with a temperature between 5 × 103 and 1.5 × 104 K (hot component) and the second component is characterized by a temperature around 1-1.5 × 103 K (warm component). The warm component dominates the emission at very late epochs (≳ +400 d), as is evident from both the last near infrared (NIR) spectrum and MIR observations obtained with the Spitzer Space Telescope. Using the BB fit to the SEDs, we constructed a bolometric light curve that was modeled together with the unshocked CSM velocity and the shock velocity derived from the Hα line modeling. The circumstellar-interaction model of the bolometric light curve reveals a mass-loss rate history with large values (1.7  ×  10-2  -  0.15  M⊙ yr-1) over the ∼25-40 years before explosion, depending on the radiative efficiency and anisotropies in the CSM. The drop in the light curve at ∼350 days and the presence of electron scattering wings at late epochs indicate an anisotropic CSM. The mass-loss rate values and the unshocked-CSM velocity are consistent with the characteristics of a massive star, such as a luminous blue variable (LBV) undergoing strong eruptions, similar to η Carinae. Our analysis also suggests a scenario where pre-existing dust grains have a distribution that is characterized by a small covering factor.