In the case of an exponential function, the amount of money he has would have to influence the amount of money he makes. For example in a savings account, the more money you have in it, the more interest you'll get (you'd get more interest on a million dollars than a hundred dollars). Thus, it would grow exponentially.
In the case of a linear function, the amount of money he has would be directly proportional to some other factor, not the amount of money he has.
In this case, the amount of money he earns is directly proportional to the number of weekends he works. If he works 1 weekend, he gets $25, 2 weekends will get him $50, 3 weekends gets him $75, and so on. If we were to plot these on a graph, we would see a straight line. So this case is a linear relationship.