Quote from Wikipedia on binary options:
Though binary options sometimes trade on regulated exchanges, they are generally unregulated, trading on the internet, and prone to fraud.[3] The U.S. Securities and Exchange Commission (SEC) and Commodity Futures Trading Commission (CFTC) have issued a joint warning to American investors regarding unregulated binary options.[4] and have forced a major operator, Banc de Binary, to cease operations in the US and pay back all customer losses. Many binary option "brokers" have been exposed as questionable operations. With such binary option brokers, there is no real brokerage; the customer is betting against the broker, who is acting as a bucket shop. Manipulation of price data to cause customers to lose is common. Withdrawals are regularly stalled or refused by such operations.[5]
I should say the same thing as /u/atcoyou. I don't know much about binary options. From what I briefly read on some random site offering binary options trading... You one of two bets, either price will go up or price will go down. It is a 50/50 bet with an all-or-none payout. Meaning if you buy a call you bet the price will go up during the stated period of the option. If the price goes up... you get your payout. If the price goes down... you get $0. I didn't register or anything but it didn't seem like there were strike prices or targets and time frame was unclear --looked like you are just betting if price will go up or down.
Getting back to your question of "what happens when you click on the call button"... well... who knows... your guess is as good as mine. According to the link and text I posted above, binary options are generally exist in an unregulated space and are offered by "brokers" with no licenses or regulatory oversight. It is the equivalent of a pure gamble. There could be absolutely nothing behind them (or the button) or it could be as simple as a bookie's statistics program similar to what you would find on calculating odds for sports betting.
Actually... after all that... the Wikipedia page actually gives you exactly what you are asking for. There are Black Scholes valuation methods for binary options posted right there on the wiki... Hope you are good with math and know the Greeks...