The igniting event is the English capture of Kate. This happens not long after she and Tom spend a great night in bed together, which strengthens the feeling - well, it's practically love - the two have for each other. Tom thus has every motive to do what he can to rescue her.
So does Blackbeard. Kate knows all kinds of things, and, as reliable as she is, Blackbeard knows that she won't be able to withstand the English torture forever. Blackbeard thus has as his highest priority the rescue of Kate - but why does he embark on the mission with a scant crew, leaving his second in command behind, Charlie Rider, while taking Tom?
Blackbeard has already shown himself to be the smartest guy around, and he demonstrates this again with a cool fake-out of the Jagger and the Brits. But in some ways even more interesting is what Tom does at a crucial moment: he kills one of the redcoats, after he's down. In other words, Tom kills one of Jagger's men, even though Tom is working for Jagger.
There are several possible motives for Tom's action. He needed to maintain his cover for Blackbeard - but couldn't he have knocked his victim out, with Blackbeard none the wiser? Maybe Tom is so deep undercover, he's actually now feeling more a part of Blackbeard's community than Jagger's? Definitely a factor. But likely the most compelling reason for killing the Brit was Tom's fury over what they were doing to Kate.
Emerging love is a powerful force. But, interestingly, Kate is a woman who loves two men - Tom and her husband. And, while we're on the subject, Selina now has something going on with both Blackbeard and Charlie Rider - not love, but Blackbeard won't be happy about it. Crossbones brews with all manner of appealing conflict.
See also Crossbones: Slow Start but Possibilities ... Crossbones 1.2: Wheels within Wheels
more ancient than Crossbones, and even more erudite