The reason I believe it is important to have an apologetic tone is to ensure you are communicating to the user that, though a mistake has been made and he is interacting with a machine or application in this case, you still respect his action and are humanizing the mistake.

To quote this article from UXMatters:

“You’re going to display your error message to a person, so write it in the tone of voice you would use if you were telling the error message to the person directly.”

I also recommend looking at this excellent article, Are You Saying “No” When You Could Be Saying “Yes” in Your Web Forms?:

Error messages seem like an unimportant and incredibly boring part of crafting a user experience. But the tonality of error messages can swing the experience around from an almost certain abandonment to a conversion.

The article emphasizes that error messages should carry a positive tone and hence an apologetic response can help users quickly recover and not immediately become defensive about a mistake they might have made.

Writing error messages that carry a positive tone isn’t rocket science. Just follow a few simple rules: First, get rid of tech lingo such as “incompatible.” Most users won’t know what this means. Speak to them in their own language and not the language of your developers. For example, “incompatible” translates to “does not work together” in plain English. Don’t use negative words Clearly identify the error so the user knows what to correct. Give the user a hint of how the problem can be solved. Put the blame on yourself, not on the user.

I also recommend looking at The Effect of Apologetic Error Messages and Mood States on Computer Users’ Self-appraisal of Performance for a study which analyzed the impact of using apologetic error messages on user mood states and how human-like apologetic tones are effective in computer interfaces. To quote an extract from the article

In HHI, apologies are generally used to express regret (Leech, 1983; Schlenker andDarby, 1981) or to alleviate individuals’ anger caused from their disapproval of others’ action.In other words, apologies mitigate frustration and anger when attempted interactions fail.Similarly, Nielsen (1998) argues that error messages responding to the computer user’s action should include a simple apologetic statement when the reason for the error is the limitation of the computer interface to perform the intended task. Tzeng (2006) conducted a study investigating users’ perceptions of online systems containing three different error messages,each of which includes different politeness strategies. In the study, firstly users’ politeness orientations were elicited and then participants were asked to interact with websites including pre-determined problems. When users encountered problems, the system provided certain error messages representing one positive politeness strategy (i.e. joke), one negative politeness strategy (i.e. a simple apology), and a mechanical message for the error (i.e. the page is temporarily unavailable). The findings of the study showed that users who deal with social events with polite expressions preferred to receive apologetic messages significantly more than mechanical or joke messages, and they preferred apologetic messages significantly more than those messages that are less oriented to polite expressions.

The article also calls out how an apologetic tone as opposed to a negative tone impacts the perception that users of them selves and how a positive apologetic perception can help in enhancing positive performance

De Laere, Lundgren, and Howe(1998) compared human-like versus machine-like interaction styles of computer interfaces.They did not observe a significant difference between the different styles in terms of users’self-appraisals. The findings of the study also demonstrated that negative feedback affects the self-perceptions of the participants differently than positive feedback. Tzeng (2004) examined whether apologetic feedback affects users’ performance perception in the computerized environment. This study suggested that users may not expect computers to be polite, but apologetic statements made the subjects feel better about their interactions with the program.

The research paper Computer Apology: The Effect of the Apologetic Feedback on Users in Computerized Environment also calls out why apologetic negative feedback ties into user experience from a social aspect

The use of apologetic statements with an error message contributes the human-computer interaction. If we consider that the main aim of the user centered design is to create an environment for users in which they feel themselves comfortable, use of apologetic statements in the user interface design become a very important issue. Moreover, in human-human interaction, one of the more important, may be the most, issues is to behave in a respectful manner. In most of the societies when a person does not behave in a respectful manner or makes a mistake towards the other person, apologizing is the traditional and the most effective way in order to overcome the problem. Similarly, this study shows that most of the subjects thought that apologetic feedbacks do not seem awkward to them and 95% of them receiving apologetic feedback felt that apologetic feedback seemed sensitive to them. Here, it seems that subjects find it interesting to confront with respectful behavior such as apologizing when they encounter an error caused by computers’ inability as if they encounter a problem in human interaction. The findings of this study indicate that representing the affective state of a person in the interface design is very important in human-computer interaction because people are more sympathetic to see emotional aspects in the interface such as, sensitivity, respect, and feeling of humanity. Therefore, these results might be used as evidence for the claim that computers’ offering apologetic statements to the users can substantiate the idea of real user centered design.

On the flip side Microsoft recommends using sorry or an apologetic tone only when a serious error has occurred: