How to Train Your Dragon
Meatlug was one of the dragons who was stealing the flock from the Vikings and feeding them to the Red Death out of fear. She was then captured by the Vikings of Berk and imprisoned in Viking training where vikings are trained to kill dragons.
Fishlegs takes her as his dragon after Hiccup Horrendus Haddock III (one of the franchise's two main protagonists, alongside Toothless) helps Snotlout Jorgenson train Hookfang. They help Hiccup and Toothless defeat the Red Death. Toothless later kills the Red Death, but Hiccup loses his leg in the process. Eventually, after this battle, vikings and dragons are able to coexist.
How to Train Your Dragon 2
Meatlug first appears in the film when she, along with Fishlegs, compete in the Dragon Races. Later on in the film, she and the Dragon Riders and their dragons (minus Hiccup, Toothless, Astrid, and Stormfly) are seen flying around Berk briefly.
Eventually, after Valka's Bewilderbeast is killed, Drago's Bewilderbeast takes control of all the dragons around him, including Meatlug herself, and even Toothless. However, Hiccup manages to free Toothless from the Bewilderbeast's control. Hiccup and Toothless confront the Bewilderbeast, but lose their fight and are presumed dead. However, it is revealed Toothless protected Hiccup from his icy breath, and he goes and confronts the Bewilderbeast himself. This drives Meatlug and all the other dragons out of the Bewilderbeast's control, and they all fire on him until he and his abusive master, Drago Bludvist, are forced to retreat. Meatlug soon reunites with Fishlegs and are later seen competing in the Dragon Races once again.