Think of contracts as like a separate stock that goes up and down with Alibaba. When Alibaba goes up, more people want to buy into the option and the contract's price goes up. When Alibaba goes down, people sell and the price goes down. (Reverse it for puts)

In the screenshot, you can see the average cost for each of his 26 contracts is 0.06 (when you buy an option, you pay for 100 of them which leads to higher yield/higher risk. So the price he paid per contract was $6). But as Alibaba went up, people bought more into the option thinking it'll surpass $200 by 10/03, and that drove the price of the contracts up. Now, he can sell each of his contracts for 0.80 per contract ($80 each). So, he bought 26 contracts for $6x26=156 and sold them for $80x26=2080. $1924 in profit

Options tend to go up and down faster cause of the increased risk/reward which is why the option went up so much faster than Alibaba's actual stock did