First of all, please get away from YouTube for anything not related to high school. Although there are some good videos out there, it's easy for an academic to pull out egregious errors in each video. Remember, you can try to explain to your prof how "YouTube said it was right," but your prof is guaranteed to tell you "YouTube is not grading your exam."
An ozonolysis reaction is an example of a 1,3-dipolar cycloaddition (+ reverse cycloaddition + re-cycloaddition) which may be thought of as a concerted process (all bond making and breaking steps occurring simultaneously). A stepwise diradical mechanism was once proposed, however there is too much evidence to support the cycloaddition mechanism. This reaction takes place via a six-electron aromatic transition state.
http://lennoxtutoring.com/2012/09/04/ozonolysis-mechanism/O2 is not eliminated from this reaction. Rather a reductant (Me2S or Zn) extrudes an oxygen atom to complete the process.
Please look up 1,3-dipolar cycloadditions in a good textbook such as Vollhardt or Streitwieser and Heathcock, however a more rigorous explanation may be found here:
http://www.chem.wisc.edu/areas/organic/studsemin/wilde/wilde-sem.pdf