Why does the human body express sadness through tears?

To explain this kind of thing, an evolutionary perspective is needed. Nature don't follow an "intelligent" design but it develops in ways that work on top of things that worked before. So, a useful question to ask would be "why does crying work?".

Crying usually happens when we need help from others. On those occasions, the emotional energy goes to places where it may activate mechanisms to symbol or ask for help; the throat and the face. Humans, unlike most animals, rely heavily on the sense of sight for many activities, including communication. If you think about it, we have an incredible capability to perceive the tiniest detail on face expression and we even perceive many things subconsciously, like the dilation of pupils. So, it makes sense that our responses are different from other animals, focusing on messages that are face to face.

Emotions are divided into two systems, one for fighting and running away and the other for resting and digesting. Crying is part of the second group, probably because we are able to suppress strong emotions while we are in the presence of a threat. That system is also in charge of excreting waste and salivating. So, when we are charged with emotions, the body activates the function associated with cleaning and healing, which for the eyes is very close to using tears to get rid of harming stuff in the eye, and it happens to also function as a very clear message to our surroundings.

It seems like crying with tears might not be a perfect message of emotion because it seems like it's wasteful since tears might not actually accomplish it's practical purpose; but, again, the body doesn't have to make sense, it only has to work under the evolutionary rules; and it does it in ways that may seem illogical using our linear thought process but it's actually much more complex because it can naturally comply with an infinite amount of variables.

In other words, tears manage to deliver the message clearly and it's easier for the body to waste some tears than to rely on any other system at hand.