I am having some difficulty understanding 'hedging a call option bought' from the following passage. What I understand from it is: in order to hedge the option we bought (i.e. the remove risk associated to the case in which the price of the option will be such that we won't exercise the option) then borrow some money from the bank and sell stocks? However, I can't see why this means that we are removing risk associated to the option.
It's probably easiest to understand this concept in the idealized world of a binomial tree where a stock can only move up or down by a known discrete amount each time step (though you don't know which way it's going to go).
Say you have a European call option that expires the next time step. Then since there are only two possible values the stock can take, there are only two possible values your option will expire with. We can call them $C_u$ and $C_d$ where the $u$ correponds to the situation where the stock goes up and $d$ to where it goes down. For instance if your stock may be worth $S_u = 20$ or $S_d=15$ the next time step and you have a call with strike $15,$ then $C_u = 5$ and $C_d=0.$
The neat thing is that we can replicate this payoff with a portfolio consisting of the stock and cash. Say we buy $a$ shares of stock and borrow $b$ dollars. If on the next time step, we sell the stock and repay the loan, our cash flow is one of two things: $$ aS_u-b(1+r)$$ or $$aS_d-b(1+r)$$ depending on which way the stock moves. The $(1+r)$ is there cause we have to pay the loan back with interest. We will simplify things and take $r=0.$ If we want this to be equal to the payoff of the option, we just set $$ aS_u-b(1+r) =C_u \\aS_d-b(1+r) =C_d.$$ Plugging in the example numbers I gave gives $$ 20a -b = 5 \\ 15a-b = 0$$ which can be easily solved to give a=1 and $b=15.$ So if we buy one share of stock and borrow $15$ dollars, we get the exact same payout as the option.
Let's say right now, the stock is worth $17.$ Then our cash flow today from buying one share and borrowing $15$ dollars is $17\cdot 1-15 = 2$ (since we're borrowing $15$ we can use those dollars toward the stock purchase and only need to come up with $2$ "out of pocket").
To the point of your actual question, we can use the replicating portfolio to exactly cancel out the risk of owning the call option. Here we have to short the replicating portfolio, which means taking the exact opposite position. Instead of buying two shares and borrowing $15$ dollars, we sell two shares and lend 15 dollars.$^*$ If buy the call option and short the replicating portfolio, then the payoffs the next time step cancel out exactly. Since you pocket $2$ dollars right now from shorting the replicating portfolio, this will be a riskless profit for you provided you pay less than $2$ dollars for the call option. On the other hand, if you can sell a call option for more than $2$ dollars, you should do that, cause you can buy the replicating portfolio for two dollars and completely cancel out the risk, and then pocket the difference. This shows that you need to charge at least two dollars if you're selling the option, and shouldn't pay more than two dollars for it. This gives us our option price. For a given binomial model, we can start one time step from expiration and work our way backwards to get a price at every node, and finally end up with a price right now.
The binomial world may strike you as overly idealized (and it certainly is, at the end of the day) but will seem at least a bit more plausible when there are a lot of nodes. The Black Scholes model that your book is presumably talking about takes a limit of this to continuous time, where the stock follows a geometric Brownian motion.
In "real life" it's also good to think about things working backwards from expiry. If your option is deeply in the money (i.e. the stock price is much greater than the strike price) a day before expiry, its incremental payoff will be pretty much the same as the stock. It's not very likely it will go out of the money. So you should hedge it with a share of stock whose incremental payoff will exactly cancel out. Your hedge ratio, usually called "delta", is one. On the other hand, if it's far out of the money, it will almost certainly expire out of the money. There is no risk there (even if there's no reward) so you should not have a position in stock and your delta is zero. If we're in the money, but further away from expiry, then the delta is between these two extremes. As time passes, and especially if the stock price increases further, you will need to sell more stock to be appropriately hedged. If the stock price decreases you will wind up needing to buy back some stock to remain properly hedged. Eventually if we get to expiry and the option is worthless we will have sold all our stock and wind up flat. Since we're buying low and selling high, this strategy of keeping our portfolio riskless will make us money. This is effectively how we value the option position: it's the money you make or lose dynamically hedging.
Just two notes to close with: First, it's not obvious that the amount of money you make is a deterministic quantity. In real life it's not but in the idealized world of Black Scholes with constant or deterministic volatility and continuous hedging, it is. And it is in the binomial world, as we've shown. Second, note that the more rapidly the stock price moves up and down, the more money we make dynamically hedging our call and the more the option is worth. So since the price risk can be more-or-less hedged away as described above, an options position is really about being exposed to volatility. One could also argue from a purely expected value perspective that options prices should be increasing in volatility, so volatility is important even if you can't hedge.
$^*$(You mentioned confusion over short selling in the comments and hopefully my reply helped. In our idealized situation here, we imagine we can short-sell costlessly. We also assume that we can borrow and lend at the same interest rate. This means everything is perfectly symmetrical... we can have a positive or negative position in stock or cash and a negative position behaves exactly the same as a positive position. It might be helpful to instead of stock, imagine our underlying is a future where a short position is a more straightforward thing. Most options in the real world are on futures anyway.)