The quickest way to do a question like this if you get it on a test is probably just to go through the answers. I'll try to rephrase the given explanation somewhat, but it's difficult to say it too differently.

1 - the only number less than 144 than has 143 as a factor is 143 - the question asks for a common factor of TWO DIFFERENT integers
2 - same rationale as above - 142 is the only number less than 144 with 142 as a factor.
3 - same rationale as above, but it's trying to trick you. 72 is the only number less than 144 with 72 as a factor, but you might want to select this one because it's also a factor of 144. However, remember that the two integers must be LESS than 144.
4 - 71 is a factor of 71 and 142. 142 is less than 144, so 71 is the answer.

144 can be expressed as 1*144 or 2*72.
If it was 144, 72 could be a divisor.
Since we are looking at numbers that are less than 144, any number close to 72 would be the answer. From the answer choices 71 is close. So the answer is 71.