The krogan hail from the planet Tuchanka in the Aralakh (the name of their star, translated as "Eye of Wrath") system of the Milky Way. From time immemorial, war and bloodshed has been a natural way of life for them. They shared their planet with fierce predators and had to fight tooth-and-nail to keep their settlements safe from both the animals as well as rival clans. Before they invented gunpowder, the leading cause of death of Tuchanka was "eaten by predators". Afterwards, it was "death by gunshot".
When they entered the atomic age and developed weapons of mass destruction, a massive planetary war among clans began, driving Tuchanka into a harsh nuclear winter. Built from birth as they were to survive in the harshest of conditions, the krogan could still live and breed, but the wars had destroyed virtually every major city and caused untold casualties. For centuries, they lived in the ruins of what was once a great civilization, and their culture entered a slow decline.
Around 1 CE, an expedition from the Citadel Council led by the Salarian Union activated a dormant mass relay, setting into motion a chain of events that led to a hostile race of spider-like aliens known as the rachni developing spaceflight technology. The rachni used this technology to try and wipe out the other races and take over the galaxy. The Council races, then consisting of only the asari and the salarians, attempted to fight off the invasion, but were overwhelmed by the vast and fast-breeding rachni armies. Attempts to negotiate with the Rachni Queens proved impossible, as the queens dwelled deep within underground caverns on the toxic rachni homeworlds.
The war continued for decades, with the Council races slowly losing ground. In 80 CE, the tide was turned when the salarians discovered Tuchanka and made first contact with the krogan clans. A program of krogan "upliftment" was approved by the Council, and the Salarian Union then relocated the surviving krogan to a more habitable planet, offering them advanced technology and spaceflight capabilities in exchange for their aid in battling the rachni. The krogan agreed, and were deployed against the Council's enemies. Their rapid breeding cycles gave them the numbers required to contend with the rachni armies, and their innate physical resiliency allowed them to survive and fight on the toxic rachni worlds. One by one, the krogan attacked the rachni planets and detonated powerful bombs within the caverns of the Queens, crippling the rachni ability to replenish their forces. In 300 CE, the rachni were officially declared extinct, and a grateful galaxy invited the krogan to establish a presence on the Citadel and start their own embassy.
As a gesture of gratitude for their services during the Rachni Wars, the krogan were allowed to colonize many worlds to accommodate their fast-growing population. However, over time the warlike tendencies of the krogan began to emerge, and greedy clan chiefs started leading expeditions to annex territory from the other Citadel races. As the other races scrambled to protect their territory, the krogan only grew more aggressive in their colonization campaign. Things came to a head when the krogan attacked and attempted to annex Lusia, an asari planet. When asked by the Council to stop, the krogan refused, and dared the other races to try and stop them. This started off the galactic war known as the Krogan Rebellions.
Having learnt vital lessons from the Rachni Wars, the asari and salarians had set up several defensive measures in case they ever went to war with a hostile race again. Powerful computer viruses cut off inter-krogan communications across the galaxy, while spies sneaked onto the warships of krogan chiefs and detonated warp bombs to methodically wipe out the enemy's leadership. However, the ability of the krogan to breed and quickly replenish their armies was an advantage that was too much to overcome.
Mirroring the events of the Rachni Wars, the tide was turned when the asari discovered the planet Palaven and made first contact with the turians, a powerful race of soldiers from a deeply militarized culture, who had already acquired spaceflight technology and colonized several uninhabited worlds in their system. Upon hearing of the krogan threat, the turian government (known as the Hierarchy) agreed to help, and threw their considerable armies and technological might into the war. The krogan were taken aback by the aggressive tactics of the turians, a sharp contrast to the relatively softer ways of the asari and salarians. The organized, highly trained turian soldiers began to dominate the krogan warriors, and the warlords grew desperate. They began bombing turian worlds with asteroids, leading to three whole planets becoming completely uninhabitable. The krogan leadership hoped that such a devastating loss of life would demoralize the turians and turn public opinion on Palaven against the war, but they underestimated the strength of the turians' warrior ethic. From birth, every turian is raised to think of the collective good before their own, and every turian soldier swears to sacrifice his life for the Hierarchy if necessary. Rather than souring public opinion, the krogan attack only made the turians all the more determined to crush the krogan once and for all.
The war dragged on for decades, with both sides being evenly matched thanks to the krogan ability to breed faster than any of the other races and thus maintain vast armies. However, the final nail in the coffin of the krogan invasion was hammered in when the salarians developed and released a bio-weapon called the genophage, which corrupted the genetic code of the krogan and left 99% of krogan females unable to carry a child to term. The krogan armies were devastated by this setback, and never recovered. Following that, they were quickly routed by the Hierarchy's fleets. The surviving krogan returned to Tuchanka and isolated themselves from the rest of the galaxy. The centuries-long lifespan of the krogan kept natural death rates low, but without the ability to produce new krogan, they became, once more, a race in decline.