Sealioning is a type of polite, persistent trolling, in which the sealioner endlessly asks, in a civil manner, for more information and evidence.
The term comes from a strip of the webcomic Wondermark by David Malki. In the comic a character in passing mentions that she does not like sea lions, only to have a large sea lion confront her and demand reasons for her dislike... following her home and politely pestering her night and day in an honest attempt to understand her prejudice. The original comic can be found here.
This is a common enough sort of trolling, but no one had come up with a good word for it yet -- so they made one up. Sealioning is any case in which a person claims an honest desire to understand another person's viewpoint, but is overbearing in their insistence that the person single-handedly explain the issue in detail, with no real evidence that they are learning anything.
This is perhaps a comparatively new form of trolling, because in Real Life the person asking pestering questions is then required to sit around and listen to the answers. Text-based communication allows one to skim and respond without having to be present for the time of composition, without fully processing the response, and often without social consequences for appearing dumb. If you find yourself talking to a sealion, the internet solution is to send a link to further reading, with a polite comment along the lines of "this explains it much better than I can". Then be ready to ignore them, because if they are really a sealion, that won't work.
The term first gained popularity during GamerGate, in which this tactic was frequently used to troll and waste opponents time and energy. It has since gained fairly widespread usage among people who spend way too much time on the internet, to the point where accusations of sealioning can be used to trollishly dismiss valid questions.
A more extreme form of badgering with accusing questions is sometimes referred to as monstering.