You are at the beginning of a period and the stock price, worth $S$, can evolve in either of the 2 states: $S_u = u S$ or $S_d = d S$.
The part you don't understand is related to forming so-called replicating portfolios. More specifically, using only the stock and a (risk-less) cash account, the question is 'How can one build a portfolio allowing to perfectly replicate the option's behaviour over a given time frame' (here a period of the binomial tree).
Let $\Pi$ denote such a portfolio. Let $\Pi$ consists of $\alpha$ shares of the stock and $\beta$ in cash. Both $\alpha$ and $\beta$ can be positive or negative depending on if you own/sold shares or borrowed/lent cash.
$$\Pi = \alpha S + \beta$$
At the moment we don't know $\alpha$ nor $\beta$. But we can determine them easily. Indeed, at the end of a period, suppose that the option is worth $V_u$ in the up state and $V_d$ in the down state. Then, because we want our portfolio to be replicating, we want its value $\Pi$ to evolve to the exact same states i.e. $\Pi_u = V_u$ and $\Pi_d = V_d$. This yields two equations:
$$\Pi_u = \alpha S_u + \beta (1 + R) = V_u$$
$$\Pi_d = \alpha S_d + \beta (1 + R) = V_d$$
The shares' component of portfolio $\Pi$ evolved from $\alpha S$ to $\alpha S_u$ or $\alpha S_d$, because the stock price evolved.
The amount of cash $\beta$ borrowed/lent at the beginning of the period, has cost/earned us some interest, hence the factor $1+R$.
Solve these 2 equations for the 2 uknowns $\alpha$ and $\beta$ to end up with:
\begin{align*}
\alpha &= \frac{V_u - V_d}{S_u-S_d} \\
\beta &= \frac{1}{1+R} \frac{u V_d - d V_u}{(u-d)}
\end{align*}
This gives you the number of shares ($\alpha$) you need to buy/sell and cash ($\beta$) you need to borrow/lend to perfectly replicate the option. Of course because the portfolio is replicating $\Pi = V$ the option value at the beginning of the period, which is a good way to check whether you made a mistake.
If you've built your option tree in the right way (which seems to be the case since you agree on the option premium) then on the first period you can compute $\alpha$ and $\beta$ from the above equations and it should give you the numbers that puzzle you, i.e. $0.8081$ and $-74.05$ respectively. Note that if you do the computations $\Pi = 0.8081\times120 - 74.05 = 22.92 = V$, hence ok.
Now you can repeat the calculation for any future period to get the amount of shares and cash you should borrow/lend to further replicate the option.
A subtle point here: the exercise considers that you are hedging the option. In other words, you are long the option but short the replication portfolio. This is why although you'll find $\alpha = +0.8081$ (buy shares) and $\beta=-74.05$ (lend cash) over the first period, you should actually reverse that position (because you are short the replicating portfolio when hedging a long option position).
