If according to physics, we cannot touch anything and the sensation appears only due to a repulsive force between the atoms, then how does friction occur?
The most fundamental explanation - the electromagnetic force. Atoms trying to push against other atoms. And either the push or pull is deflected or shears off atoms molecules or groups of molecules leading to energy loss, through a greater degree of'jiggling' of the atoms in the vicinity of the frictional interface (increase in temperature). And so then this energy eventually lost to radiation or conduction.
Engineers and I suppose physicists also categorize friction as static or dynamic according to whether the object or system experiences friction in a non-moving or moving state respectively. In static friction the two members are locked in equilibrium, the applied force being countered by opposition forces at the atomic level. This state can only exist up to a threshold of applied force after which motion happens and you have dynamic friction.