Relationships of all kinds should be built on respect and trust. A vital part of maintaining that level of respect and trust is being accountable for your own actions. The problem is that society has promoted this idea of fun-size apologies that act as a portion of a checklist rather than an indication of growth and maturity.
While apologizing for bumping into someone is different than apologizing for something big, those big apologies require active participation. Apologies aren't easy, and they shouldn't be. They are a way for people to recognize that their behavior has to change.
The trend of passively saying "I'm sorry" as a way of apology isn't fair to anyone within the situation. The thing is, proper apologizes should lead to a conversation, and changed behavior. There are two parts to every conversation: talking and listening. If someone is trying to apologize without listening, this means that there is no longer a conversation. It is a confrontation.
The second important part of an actual apology is making an actual effort to change the behavior that caused the issue. If the person apologizes but refuses to use this as a learning experience, the apology wasn't sincere. The problem with problematic behavior is that people have to actively break the habit of that behavior. Apologies without growth show that the person apologizing is doing so for selfish reasoning.
Apologies aren't for the person saying sorry. They are for the person who was impacted negatively. If the person who did not display troublesome behavior has to seek out or ask for an apology, then the person apologizing needs to pay even more attention to the behavior after the fact. If you are unable to recognize the problem you may have caused, listen carefully and critically apply what the person is saying.
The art of saying "I'm sorry" is critical to forming a long-lasting and healthy relationship. Accepting that you were in the wrong helps to build maturity and accountability. It's your behavior, make sure you own up to it.