In my opinion, stereotypes affect our interactions with people from other groups because we (consciously or not) treat them as if the stereotypes about their group were true. This harms our ability to interact with them as individuals rather than as members of their group.
For example, when we interact with old people, we may find ourselves treating them as if they were feeble in both mind and body. Or when we interact with students of ours who are, say, athletes or cheerleaders, we might treat them as if they are shallow and stupid. (These will depend on our stereotypes of the given groups.
This hurts us by denying us the chance to get to know the individuals rather than the stereotypes. It also harms our interactions because it surely annoys the people we are dealing with and makes them think much less of us for treating them in these stereotyped ways.