It is a decision problem, it is always a decision problem...
The most basic decision problem is a lottery that gives you X (X > 0) for a chance of %Y and nothing for %(100-Y). Someone comes and offers you to buy your ticket for Z (0 < Z < X, otherwise it is trivial). What would you do? If Z is close to X, you need to be more risk-willing (or seeking) to refuse the deal. If Z is close to 0 you need to be more risk-averse to accept the deal. Usually if Z is somewhere close to X*Y/100, you are called risk-neutral. The function that "regulates" your responses to such situations is called your utility function (curve). In the most basic terms it is between risky asset (i.e. a stock) and a risk-free asset (bond).
The problem is then modeled as an equilibrium to find the "fair price". There is usually an implicit assumption on your utility function (some diminishing returns curve, less and less marginal utility for marginal dollar gains). Fair price can be defined as "a point where no party gets a better deal than the other or by making a deal with a 3rd party". Because if it would make me worse on a deal, I would not agree. And if I can make a better deal with a 3rd party I would take that deal. If the deals are not correctly priced, they can be constantly abused (it is not arbitrage, it is the law of large numbers). The fair price means you are indifferent to taking any side of the bet or investing in risk-free rate.
In the case of positive risk-free rate, a third option arises; keep your money to yourself. Though, negative risk-free rate occurs in real life. So, there is more to life than trading (probably keeping large amounts of money safe might be more expensive than the risk-free rate provided).
There is a positive risk free rate because there exists bodies that are willing to take the risk and believe they will make more money than they offered to you.
The question "is there really a risk-free rate?" notwithstanding, why would you choose to hand over your money for a fixed return if you can access to the same deal the person will make with your money? Suppose there are infinite deals and all probabilities and payoffs are well defined. Would utility curves matter? The only thing that would prevent utility curves to be obsolete is the absorbing state (losing all your money). You cannot return from that. Assuming not playing is not an option, eventually there will be a single winner with all the money and all others broke.
In real life, time is also a resource so the economy expands, rules are more complex and probabilities not clear. As a result, utility curves are various and changing. There is also consumption. So positive interest rate is a belief for the offering party they can do better performance with that money of yours and an incentive for you to avoid spending that money or investing in something else.
No free lunch is something associated with infinite abuse of the contracts.
And a technical reason for options pricing, keeping risk free rate positive helps some models' properties to stay same. For example if your risk free rate is negative it might be justified to exercise a non-dividend paying stock's american option early and not justified to exercise the put.