The example the reference gives you doesn't make sense. To give someone the axe means to take away someone's position or job. And it can also mean stop the progress of something. eg. The company I worked for was sold and I got the axe! (Its an idiomatic way of saying lose your job.) eg. The funding for the research project got the axe. The funding stopped.
Over, in the sentence means in favour of Rob or 'why did you give me the axe instead if Rob'.
Hope that helps, Helen.