Say a stock is at $10. You buy an option, which is a contract, that allows you to purchase that stock for $15 anytime within the next 3 months(which is when the option contract expires).
Now one month later the stock is at $20 per share. Your option you bought that allows you to purchase the stock for $15 (which is called the strike price) is still good of course. You may now go purchase 100 shares (a standard size for 1 option contract) at $15 a share and immediately sell all of those shares for $20 a share. Instantly making $500 net.
In practice, most option contracts hit their expiration and are worthless. They are "out of the money" options. Basically if that stock went down to $8 per share, you would not "exercise" your option to buy it at $15, because that would be more expensive than market price...and you would let it just expire.
There are a ton of strategies with options, from highly leverage speculative investing to using them as tools to hedge positions.
So if somebody bought 1700 call options right before, at which time they were "out of the money", and then sold them at the peak twitter high price of today...they would have made around $255,000 if they choose to exercise the options. But that's assuming they bought at a strike price that was the same as the market price at the given time...around $37/share. All that money for a relatively low upfront investment risk.
The 1700 calls could be related, or could have been stupid luck. Call options are fairly common and that person did it at the right time. The real question is, did they exercise the options or sell off the option contracts (which is what is done most of the time) and actually make all that money. If so, there might have been a little more going on than dumb luck (fraud).
The days of hiding in the Caymans or in Switzerland are over.