Your teacher is talking about the conversion factor to convert moles C6H12O6 to moles C. That is how you isolate C from the whole molecule.
There are 6.02 x 10^23 molecules (particles) of C6H12O6 in a mole of anything (in this case a mole of C6H12O6). So how many molecules are in 4.25 moles. That must be obvious that it is 4.25 x the number in a mole but to do it with conversion factors works like this.
4.25 moles C6H12O6 x (6.02 x 10^23 molecules C6H12O6/1 mole C6H12O6) = 2.558 x 10^24 molecules C6H12O6. Some call this the unit factor method and some call it dimensional analysis. At any rate, note that the factor is placed (it could be upside down if necessary) so that the unit we don't want (moles in this case) cancel and the unit left standing is molecules (particles) and we want the answer in molecules. So far so good BUT we have the number of molecules in 4.25 moles C6H12O6. So now we use another factor to convert from moles C6H12O6 to moles C.
2.558 x 10^24 molecules C6H12O6 x (factor) = ?? moles C
2.558 x 10^24 molecules C6H12O6 x (6 molecules C/1 molecule C6H12O6) = 1.535 x 10^25 molecules C which I would round to 1.54 x 10^25. Note again that molecule C6H12O6 cancel and leaves standing the unit we want which is molecules C. Please respond if this is not clear but explain what you don't understand about it. This could have been done in one step by using the two factors at the same time; ie.,
4.25 moles C6H12O6 x factor x factor.
4.25 moles C6H12O6 x (6.02 x 10^23 molecules/1 mole C6H12O6) x (6 molecules C/1 molecule C6H12O6) = 1.54 x 10^25 molecules C.
Check my thinking. Check my arithmetic.