The mutant dinosaurs are Stromlings
We have no confirmation about the backstory of these mutant dinosaurs... all we know for sure is that they came the heck out of nowhere and wrecked LEGO City. This takes place in 2010, in-universe time. In real time, 2010 saw the release of LEGO Universe (and depending on whom you ask, the game's events might have taken place in 2010 too). LEGO Universe introduces the Maelstrom, a malevolent force of chaos that mutates LEGO denizens into nightmarish monsters to spread destruction and ruin across the universe. Seeing as the Dino Attack dinosaurs are mutants with awesome abilities that seem to exist for the sole purpose of leaving LEGO City in post-apocalyptic ruins, it's possible that these dinosaurs were mutated by the Maelstrom. In that regard, while Stromling Pirates, Admirals, and Apes may be Gnarled Forest's unique Stromling breeds while Dark Ronin, Maelstrom Horsemen, and Maelstrom Dragons are Forbidden Valley's unique Stromling types (and so on and so forth), the mutant dinosaurs may be considered Earth's own Stromling breed.- And yes, this is the premise of the Adventurers' Island Story Arc of Dino Attack RPG.
The dinosaurs are cyborgs
There's a fair bit of discussion about whether the dinosaurs are robots or biological mutants. But what if they're both? While we have no idea who created them or why, it could be that they were not only given genetic mutations, but also mechanical enhancements (like the "internal furnace" mentioned on the official site's page for the T-rex).