The Unity Engines are a key component in Sigma's plans for the Worlds Unite story. With them, Sigma planned to channel the energy of the worlds into himself, furthering his twisted ideas of evolution and ultimately to ascend to godhood.
The purpose of the Unity Engines is two-fold. First, they open Genesis Portals to bring the worlds in contact with each other. Second, they begin fusing the planets together, allowing Sigma to siphon their energy longer without destroying them in the process. These are, in turn, channeled through a Master Engine which sends the energy into Sigma himself, granting him new powers the more he absorbs.
Sonic Man and M'egga Man were supposed to oversee the arrival and activation of the Unity Engines on the worlds of Mega Man and Sonic, respectively, but interference prevented them from doing the latter right away. Ultimately, they were activated and the planets were fused together, although this also activated a secondary programming secretly implemented by Wily and Eggman that had Sonic Man and M'egga Man fighting each other as soon as they met in an attempt to sabotage Sigma's plans.
Later, after the Unified Army were worn down by their attempts to stop Sigma at the Lost Hex, Sigma used his resurrected Maverick army to send dozens of Unity Engines to multiple worlds in order to further increase his powers. While the army was able to nullify several of them and recruit many heroes to aid them, many worlds were still afflicted with the engines, causing Sigma to grow into an even more powerful form. But in his arrogance, Sigma left the Master Engine which channeled all his new power unguarded, allowing Dr. Wily, Dr. Eggman and Xander Payne to reach it without problems. The mad doctors began attempting to use the stored energy for their own plans, weakening Sigma enough for Super Sonic and Super Mega Man to kill him. Xander Payne would then use this energy in combination with his Genesis Portal creation ability to travel through time and stop Sigma from invading Sonic's world, undoing the events of the story.