Thematically, it forms a duology with Mutineers' Moon; the latter dealt with the suppression of Anu's mutiny as part of the groundwork for repelling the Achuultani assault, whilst Heirs of Empire is more of a stand-alone bildungsroman work concerning survival on a remote planet.
As the highest-ranking officer of the Imperium present, MacIntyre had elevated himself to the rank of "Governor of Earth" in order to absolve the loyalist mutineers of their crimes; he then unified the worlds' governments under his authority (backed up by his advanced Imperial armaments and Dahak) and set Horus the task of preparing defenses against the Achuultani scouts, which have been methodically advancing on the Sol system, detected by self-destructing Imperial sensor arrays.
This gave frantic defenders under Lieutenant Governor Horus and his assistant, Gerald Hatcher, barely two years to pacify the holdouts to the new world order (such as the Asian Alliance), to modernize the world economy, and to construct and power a planetary defense shield, as well as construct and train a space fleet and the fortresses on the ground which will support the fleet; and then of course to defeat both the scouts and the main Achuultani incursion.
Book 2 begins with a different point of view; the subject is now a minor Achuultani tactical officer named Brashieel, attached to the scout forces about to drop out of hyperspace and destroy Earth.
The ambush, while successful (because of the element of surprise and the generally superior Imperial technology), nevertheless sets the tone for the rest of the Siege by being extremely bloody on both sides.
There they briefly engage the Achuultani (to ensure they are sucked far enough into the system, past the hyperlimit that they cannot escape, and to gather some more military information); the opposing fleet's having stepped into Colin's "mousetrap", the Enchanach Drives of eight planetoids simultaneously activate, inducing a supernova which obliterates Sorkar's forces.
Like in the first, the Guard ambushes the second force (this time laying a dense field of hypermines, which account for a quarter of the million Achuultani vessels), but some of Sorkar's couriers had escaped and warned Hothan's fleet of the nova trap, so that stratagem was unusable.
Instead, Colin traps Hothan's forces in normal space (again, using the Enchanach Drive's side effect to exploit the hyperdrive's limitation of being unable to work in a sufficiently deep gravity well).
This is so effective because it had been previously discovered that the explanation for the various Achuultani anomalies (their lack of females, the oddly inconsistent state and stasis of their technology, their constant war making and hyper-xenophobia etc.)