Both positive and negative diverge on their own. That's the crucial part. Suppose I have 1+1+1+... and -1-1-1..., I can rearrange it to be "more positive" by doing 1+(1-1)+(1-1)+... to get 1, or 1+1+(1-1)+(1-1)+... to get 2, or 1+1+...{k times}+(1-1)+(1-1)... to get arbitrary positive k. Of course this is not an example of conditionally convergent series but it illustrates the point of what you can do when each positive and negative parts diverge.