By using a capital letter and thus personifying Nobody, the speaker is actually acknowledging that somebody is a living entity. By asserting that she, the speaker, is "nobody" using this personfication, she is actually achieving the opposite.
Of couse this irony is not an accident. The point that the speaker is making is that nobody is Nobody. Everyone is a person of value. However, she similarly assigns a persona to Somebody in a similar manner. The irony here is that the Somebodies and the Nobodies stand in conflict. As a conflict typically has two sides, it is made more clear that the Nobody entity does, indeed, exist.
In addition, the Nobody meets another Nobody. Again, the "nobody" designation has now advanced from a single entity to a group.