Someone may have better words to explain this than me, but the big thing you have to remember is voltage is a potential difference. In most cases the "difference" part is a difference between some potential and ground potential. When someone says -5v, they're saying that you are below ground.
You also need to keep in mind that voltage is relative. So like I mentioned before, most people reference to "ground"; but what is ground? You can say ground is earth ground, but what about the case when you have a battery powered device that has no contact to ground. In this situation we have to treat some arbitrary point as "ground". Usually the negative terminal on the battery is what we consider from this reference.
Now consider the case that you have 2 batteries in series. If both were 5 volts, then you would say you would have 10 volts total.
But the assumption that you get 0/+10 is based off of "ground" as being the negative terminal on the battery that isn't touching the other battery and then 10V as being the location of the positive terminal that isn't touching the other battery. In this situation we can make the decision that we want to make the connection between the 2 batteries be our "ground" reference. This would then result in +5v on one end and -5v on the other end.
Here is what I was trying to explain:
+10v +++ +5v
| | < Battery
+5v --- 0v
| | < Another Battery
0v --- -5v