Rem left the Earth aboard Project Seeds when the planet's resources were all consumed and it became inhospitable. In their search for a new home, the remnants of humanity, with their cryogenically frozen human cargo in tow, traveled the stars and came upon a young Vash and Knives alone on a hostile planet. While most of the crew wanted to exterminate them, Rem stood in the way and saved them. She was treated Vash and Knives like human beings, and not monsters as did most of the crew. Rem's favorite place aboard Seeds was the holographic Eden that was a smaller simulation of her old home on Earth, without any war, stealing, hate. Vash grew very attached to Rem, and adopted her life philosophy. His brother, Knives, also loved her, but took a different meaning from her teachings, which some see as almost refuting them altogether.

Spoiler

As Vash and Knives grew apart on the grounds of their disagreement, Knives developed a hatred for humanity and attempted to sabotage the Seeds mission. However, as the ships sped towards a fiery crash on the new home planet, Rem ushered Vash and Knives into an escape pod and went back to correct the trajectory with which Knives had tampered. By sacrificing herself, she succeeded in securing the survival of the human race. Rem is never far from Vash's heart and is often in his thoughts, even after over a hundred years.

Rem is Vash's inspiration. She's the reason why he won't kill, for she once had said "No one has the right to take the life of another." She's also the inspiration for Vash choosing to wear a red coat. Rem's favorite flower was the red geranium, a flower that symbolized unshakable determination. Wearing the red coat not only reminds him of Rem, but also reminds him of his promise not to kill and to help others. Source: http://www.absoluteanime.com/trigun/rem.htm