Following the The League Civil War, the newly reformed League had just recovered from a massive conflict and had lost many members. Seeing this weakness, the Reapers took their chance to strike. They sent a single Reaper to sabotage the Rachni homeworld of Suen, and 'sour' the Rachni Song (Basically how they raise their young 'uns). As a result of this, the Rachni Queens and their Hives were driven insane and decided to attack everyone.
At first, it went disastrously for the League. The Rachni had trillions of soldiers and surprisingly sophisticated ships. They took over many League planets and negotiations were out of the question, as contacting the Queens beneath their toxic worlds was nigh impossible. Not only that, but the majority of the League's army had their own problems to deal with, the Daleks were fighting a Civil War and the Elites were doing odd jobs for the Prophets.
Just when many were thinking of a mass exodus, explorers happened upon the star of Aralakh, home system of the Krogan. Still recovering from a nuclear civil war, the Krogan were in a pre-industrial state when they were discovered on Tuchanka. League races uplifted them, teaching them how to use modern technology and fight effectively. They then entered the war.
Unfazed by the toxic atmospheres of the Rachni worlds, Krogan forces fought the Rachni on every front. They were both quick breeding races, but the Krogan ability to reproduce far outmatched the Rachni's. Laying 1,000 eggs per clutch, Krogan were being replaced much faster than they were being killed and soon the tide turned.
Krogan began to evict Rachni from League worlds, demolishing their underground cities and decimating their troops in ground warfare. Eventually, the Rachni were pushed back to Suen, their homeworld. The Krogan showed no mercy, pursuing them underground and slaughtering them by the thousands. The Krogan detonated warheads underground and leveled cities. There was no call for surrender from the Rachni, and they fought till the bitter end until they were finally completely wiped out in 10 PLF.
For their service, the Krogan were accepted into the League and given new worlds to settle, so long as they maintain the 20-Child policy, they were free to use these worlds as they saw fit. The Krogans rapid colonization of worlds soon led to the Alamar Incident.