**Is this something that predates back many civilizations ago? Or is this a relatively newfound trend?**

In general, it is a relatively new trend of the last few centuries, and many old cultures have/had no such concept or tradition.

Keep in mind that surnames in many cultures are a relatively new trend. There was no name to drop upon marriage if you didn't have a surname. The Nordic cultures, for instance, historically used patronymic "surnames". These generally would not change. Emma Jonsdóttir does not cease to be Emma, *Daughter of Jon* upon marrying Erik Eriksson.

Similar practices existed among other peoples such as the Welsh, for example, well into the Early Modern Era. On the other hand, some cultures like Greek would have referred to the married woman as "wife of Erik".

In cultures with longer histories of using surnames, many did not historically expect women drop their surnames at all. In the far east, examples include the Koreans, Japanese or Vietnamese. While some Chinese women added their husbands' surname on top of their own, it is a social use and not a surname change. Most women in recorded Chinese history are in fact identified by their birth surnames only (i.e. `my honourable mother <Maiden Surname> from Kiangnan`).

The same is also true in the Near East, among Iranians and Arabs. Even in Western Europe, up until the the Early Modern Era, Scottish women of the lowlands did not customarily drop their maiden names. There are also the well known examples of Romance cultures such as the Spanish. Further back, Ancient Roman women did not change their [*`nomen`*](http://en.wikipedia.org/wiki/Roman_naming_conventions#Nomen) upon marriage, either.

In the English speaking world, dropping the maiden name became standard after surnames became common among the English people; sometime in the 13th and 14th centuries. So no, it did not predate civilisation. In fact, this whole practice is not nearly as common or "traditional" as it might seem. The main cultures to have such a tradition seem to be Anglophone, (Germanic) Western Europe and Slavic ones.

**Why is it that the maiden name is traditionally dropped when a woman is getting married?**

This is a lot more murky. In the English tradition, it is often said to be rooted in women being quasi-properties of their husbands without a separate legal existence; and that therefore they take their husbands name to mark themselves as extensions of the man of the family. It is difficult to determine the veracity of this claim.

Generally speaking, however, I'm more inclined to argue that the practice of dropping maiden names occur in two situations:
 
 1. In societies that did not have strong, blood-oriented views of family. So upon marriage, women are seen to have joined a different family, and the husbands' names are adopted in recognition of their new family. In contrast, culture that valued **blood** lines (e.g. Japanese clans) held on to their own clan names.
 2. Societies that did not have surnames until relatively recent, which overlaps with point 1. Cultures with strong views of family tend to adopt a collective **representative** name. In those without, surnames tend to come about for **identifying** otherwise similarly named individuals. It would have been convenient, and would indeed make sense, to identify a wife by her husband's name (*"Agnes who married John the smith; not Agnes who married John the carpenter"*). Then as surnames became more established, dropping maiden names turned into an ingrained  traditional custom.