To solve this question, we need two bits of information:
1) What a mole is
2) Given what a mole is, how to find the number of oxygen atoms
We'll start by outlining what a mole is (see link below for more info). A mole is simply put, a very large number of something. It's not just some random number, and the premise behind the mole is a big deal as far as chemistry goes. Formally, a mole of X contains 6.02*10^23 units of X. This could be cars, bananas, or parakeets, but usually, it refers to molecules and atoms.
For our purposes, the statement that we have 1 mole of Fe2O3 means that we have 6.02*10^23 molecules of Fe2O3.
Now we'll move on to the second part. There are two ways we can find the number of oxygen molecules. We'll do the systematic way first that uses proportions. I know that in one molecule of Fe2O3, I have 3 Oxygen atoms, so can set up a proportion like the one below:
3 atoms O/1 molecule Fe2O3
X atoms O/(6.02*10^23) molecules Fe2O3
We can solve this by multiplying both sides by 6.02*10^23 molecules of Fe2O3 and cancelling the units to end up with the following equation:
X atoms O = 3*6.02*10^23 = 1.806 * 10^24 atoms O
Therefore, our result is 1.806*10^24 atoms of oxygen.
The second, quicker way to do this second part is simply to recognize that based on the chemical formula that you'll have 3 times as many oxygen atoms as molecules of Fe2O3, so you might've just multiplied 6.02*10^23 by 3 to get the result.