In a one-period model, a combination of buying the underlying asset and borrowing or lending money can reproduce the two possible payoffs of an option. The cost of that combination is the option price, and the real-world probability of the underlying rising does not enter the price. In a tree with many steps, the same calculation is repeated one step at a time from the payoffs at expiry to find the price today.
Consider a hypothetical example in which the underlying price is 100 today and will be 110 or 90 after one period, and the one-period interest rate is 2%. A call with a strike price of 100 is worth 5.88. Buying 0.5 units of the underlying and borrowing 44.12 reproduces the payoff at the end of the period (10 or 0).