The actor, who plays Ben Mason, admitted to BuddyTV that the alien-invasion drama is "a slow-burn type of show".
Jessup said: "As the series goes on, every episode you learn a little bit more... about the aliens and about what they want, and especially in relation to my character."
Jessup promised that the central mystery of the harnessed children will be unveiled towards the end of the season.
"You learn what's happening with these kids and what these harnesses do and how they tie everything together," he said. "Each episode reveals a little bit more of the mystery or pulls back a little bit more of the veil, so I think that's what will keep people watching."
Jessup also suggested that the first season's ten-part run does not leave room for "filler episodes".
"There's no junk episodes," he said. "Every episode is integral to the overall plot."
Falling Skies was recently renewed for a second season, while series producer Mark Verheiden revealed that he has long-term plans for the show.
Falling Skies airs on Sundays at 10pm on TNT in the US and on Tuesdays at 9pm on FX in the UK.
> Falling Skies Colin Cunningham: 'Pope won't change'
Watch a trailer for Falling Skies below: