Recall that a covalent bond can be considered a sharing of electrons. The NH3 nitrogen used to have two electrons in its lone pair and was neutral, but it formally only has one electron in the formed N-C bond (because the other one now belongs to the carbon). A loss of one electron from a previously neutral species leads to a singly positive species. This is why you get a + on the nitrogen and it's the same thing with oxygen.
The nitrogen becomes neutral again when it loses a proton and one of the electrons which belonged to the H in the N-H bond formally transfers to the nitrogen.