We explore the cosmological implications of anisotropic clustering measurements in configuration space of the final galaxy samples from Data Release 12 of the SDSS-III Baryon Oscillation Spectroscopic Survey. We implement a new detailed modelling of the effects of non-linearities, galaxy bias and redshift-space distortions that can be used to extract unbiased cosmological information from our measurements for scales $s gtrsim 20,h^{-1}{rm Mpc}$. We combined the galaxy clustering information from BOSS with the latest cosmic microwave background (CMB) observations and Type Ia supernovae samples and found no significant evidence for a deviation from the $Lambda$CDM cosmological model. In particular, these data sets can constrain the dark energy equation of state parameter to $w_{rm DE}=-0.996pm0.042$ when assumed time-independent, the curvature of the Universe to $Omega_{k}=-0.0007pm 0.0030$ and the sum of the neutrino masses to $sum m_{ u} < 0.25,{rm eV}$ at 95 per cent CL. We explore the constraints on the growth rate of cosmic structures assuming $f(z)=Omega_{rm m}(z)^gamma$ and obtain $gamma = 0.609pm 0.079$, in good agreement with the predictions of general relativity of $gamma=0.55$. We compress the information of our clustering measurements into constraints on the parameter combinations $D_{rm V}(z)/r_{rm d}$, $F_{rm AP}(z)$ and $fsigma_8(z)$ at the effective redshifts of $z=0.38$, $0.51$ and $0.61$ with their respective covariance matrices and find good agreement with the predictions for these parameters obtained from the best-fitting $Lambda$CDM model to the CMB data from the Planck satellite. This paper is part of a set that analyses the final galaxy clustering dataset from BOSS. The measurements and likelihoods presented here are combined with others in Alam et al. (2016) to produce the final cosmological constraints from BOSS.
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