The S-wave $D{D}^{⁎}$ scattering in the isospin $I=0,1$ channels is studied in $N_f=2$ lattice QCD at $m_{\pi }\approx 350$ MeV. It is observed that the $D{D}^{⁎}$ interaction is repulsive in the $I=1$ channel when the $D{D}^{⁎}$ energy is near the $D{D}^{⁎}$ threshold. In contrast, the $D{D}^{⁎}$ interaction in the $I=0$ channel is definitely attractive in a wide range of the $D{D}^{⁎}$ energy. This is consistent with the isospin assignment $I=0$ for $T_{cc}^{+}\left(3875\right)$. By analyzing the components of the $D{D}^{⁎}$ correlation functions, it turns out that the quark diagram responsible for the different properties of $I=0,1$ $D{D}^{⁎}$ interactions can be understood as the charged ρ meson exchange effect. This observation provides direct information on the internal dynamics of $T_{cc}^{+}\left(3875\right)$.