We explore the cosmological implications of anisotropic clustering measurements in configuration space of the final galaxy samples from Data Release 12 of the Sloan Digital Sky Survey III Baryon Oscillation Spectroscopic Survey. We implement a new detailed modelling of the effects of non-linearities, bias and redshift-space distortions that can be used to extract unbiased cosmological information from our measurements for scales s ≳ 20 h-1 Mpc. We combined the information from Baryon Oscillation Spectroscopic Survey (BOSS) with the latest cosmic microwave background (CMB) observations and Type Ia supernovae samples and found no significant evidence for a deviation from the Λ cold dark matter (ΛCDM) cosmological model. In particular, these data sets can constrain the dark energy equation-of-state parameter to wDE = -0.996 ± 0.042 when to be assumed time independent, the curvature of the Universe to Ωk = -0.0007 ± 0.0030 and the sum of the neutrino masses to ∑mν < 0.25 eV at 95 per cent confidence levels. We explore the constraints on the growth rate of cosmic structures assuming f(z) = Ωm(z)γ and obtain γ = 0.609 ± 0.079, in good agreement with the predictions of general relativity of γ = 0.55. We compress the information of our clustering measurements into constraints on the parameter combinations DV(z)/rd, FAP(z) and fσ8(z) at zeff = 0.38, 0.51 and 0.61 with their respective covariance matrices and find good agreement with the predictions for these parameters obtained from the best-fitting ΛCDM model to the CMB data from the Planck satellite. This paper is part of a set that analyses the final galaxy clustering data set from BOSS. The measurements and likelihoods presented here are combined with others by Alam et al. to produce the final cosmological constraints from BOSS.