This problem assumes there are two possible states of the market in a single time period.
At time $t=0$, the asset price is $S_0 = 100$ and the call option price is $C_0$, to be determined. At time $t=1$, there are two possible asset prices $S_{1u} = S_0(1+u)=200 $ with probability $p$ and $S_{1d} = S_0(1+d)=50 $ with probability $1-p$. In this case, $u=1$ and $d = -0.5$ Let $K=150$ denote the option strike. The payoff of the option at expiration is
$$C_1 = \max(S_1-K,0), $$
where the random variable $S_1$ is $S_{1u}$ or $S_{1d}$ depending on the future state of the market:
$$C_{1u} = \max(S_{1u}-K,0)\\\ C_{1d} = \max(S_{1d}-K,0)$$
The value of the option at time $t=0$ is the discounted expected value of the payoff. However this expected value is not calculated with real probability $p$ -- but rather the risk-neutral probability $\hat{p}$ that preclude arbitrage opportunities.
We can determine the no-arbitrage risk-neutral probability $\hat{p}$ by showing it is possible to construct a hedged portfolio of the option and the asset that is risk free -- it has the same value in both future states. Hence the value of the portfolio grows in time at the risk-free rate of interest $i$.
Suppose the portfolio is long $1$ call option and short $\Delta$ shares of the asset. The value at time $t=0,1$ is
$$V_t= C_t-\Delta S_t.$$ We can solve for the hedge ratio $\Delta$ so that the value of the portfolio at time $t=1$ is independent of the state of the market:
$$C_{1u}-\Delta S_{1u}=C_{1d}-\Delta S_{1d},$$
or
$$C_1-\Delta S_1=C_{1u}-\Delta S_0(1+u)=C_{1d}-\Delta S_0(1+d).$$
This value of the hedge ratio is independent of the probabilities for the value of the asset at time $t=1$:
$$\Delta = \frac{C_{1u}-C_{1d}}{S_0(u-d)}$$
Consequently, in the absence of arbitrage, the portfolio grows at the risk-free rate:
$$C_1-\Delta S_1 = (C_0 - \Delta S_0)(1+i)$$
and we can solve for the value of the call option at time $t=0:$
$$C_0 -\Delta S_0= \frac{C_{1u}-\Delta S_0 (1+u)}{1+i},$$
$$C_0 = \frac{\Delta S_0 (1+i) +C_{1u}-\Delta S_0 (1+u)}{1+i},$$
$$C_0 = \frac{-\Delta S_0 (u-i) + C_{1u}}{1+i}.$$
Substituting for $\Delta$, we obtain
$$C_0 = \frac{\hat{p}C_{1u}+(1-\hat{p})C_{1d}}{1+i}=\frac{1}{1+i}\{\hat{p}\max[S_0(1+u)-K,0)]+(1-\hat{p})\max[S_0(1+d)-K,0)]\},$$
where
$$\hat{p} = \frac{i-d}{u-d}.$$
This has the form of an expected value with a different probability -- the risk-neutral probability. The fair value of the call option is the discounted expected value under the risk-neutral probability measure.
To understand the no-arbitrage argument, consider the fact that the hedged option portfolio assumes the same value in the future regardless of the state of the market. If this portfolio did not grow at the risk-free rate, then we could make a sure profit. For example if the portfolio grows at a greater rate, we can borrow cash at rate $i$, buy the portfolio, earn the sure payoff and repay the loan with a sure profit.
In general, for any contingent claim on an asset, the no-arbitrage fair value is the discounted expected value of the payoff under the risk-neutral measure. For this simple binomial model, we can find the risk-neutral probabilities in a more straigtforward way. The present value of the underlying asset itself must equal the discounted risk-neutral expected value of the future price.
Hence,
$$S_0 = \frac{\hat{p}S_0(1+u) + (1-\hat{p})S_0(1+d)}{1+i},$$
and
$$\frac{\hat{p}(1+u) + (1-\hat{p})(1+d)}{1+i} = 1.$$
Solving for the risk-neutral probability $\hat{p}$ we again obtain
$$\hat{p} = \frac{i-d}{u-d}.$$
