You use the no free lunch principle. You describe a strategy that will guarantee a profit if $c$ is less than either number. The $0$ comes from noting that if somebody will pay you to to take the option you can collect the payment and let the option expire. The other term comes from selling a share and buying an option, then exercising at expiration. If the option price is below this value, you make a guaranteed profit.
A good way to simplify it is to assume $r=0$, which is not far wrong today. Then it just says that if the strike price is below the stock price, the option has to cost at least the difference. If it didn't, instead of buying the stock you would buy the option and exercise it. The effect of interest is to reduce the value of the $K$ you will have to pay in the future, so the option price has to be a little higher yet to avoid this arbitrage.