At low temperatures, EuTiO$_3$ system has very large resistivities and exhibits colossal magnetoresistance. Based on a first principle calculation and the dynamical mean-field theory for small polaron we have calculated the transport properties of EuTiO$_3$. It is found that due to electron-phonon interaction the conduction band may form a tiny subband which is close to the Fermi level. The tiny subband is responsible for the large resistivity. Besides, EuTiO$_3$ is a weak antiferromagnetic material and its magnetization would slightly shift the subband via exchange interaction between conduction electrons and magnetic atoms. Since the subband is close to the Fermi level, a slight shift of its position gives colossal magnetoresistance.
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