The answer is given but I'd like to understand how one would get to the answer.

Answer:
Find the limiting reactant.
Explanation:
Using the ratio in the chemical formula, find the limiting reactant.
[tex]0.164mol Mn*\frac{2molMnO3}{2molMn} = 0.164molMnO3[/tex]
[tex]0.232molO2*\frac{2molMnO3}{3molO2} = 1.55molMnO3[/tex]
The second equation results in less mol of MnO₃, therefore it is the limiting reactant and changing the moles of O₂ to moles of MnO₃ which will give the theoretical yield which is 1.55 mol MnO₃.