The simple answer is that General Lee didn't want to see his men destroyed. There was correspondence between Generals Lee and Grant in the days before the actual surrender, as both recognized the disparity of position between Lee's Army and the larger, better supplied Union Army that kept pushing him west, away from Richmond and away from supply.
Prelude to Surrender
On April 3, Richmond fell to Union troops as Robert E. Lee led his Army of Northern Virginia in retreat to the West pursued by Grant and the Army of the Potomac. A running battle ensued as each Army moved farther to the West in an effort to out flank, or prevent being out flanked by the enemy. Finally, on April 7, General Grant initiated a series of dispatches leading to a meeting between the two commanders.
"General R.E. Lee, Commanding C.S.A.: 5 P.M., April 7th, 1865. The results of the last week must convince you of the hopelessness of further resistance on the part of the Army of Northern Virginia in this struggle. I feel that it is so, and regard it as my duty to shift from myself the responsibility of any further effusion of blood by asking of you the surrender of that portion of the Confederate States army known as the Army of Northern Virginia. U.S. Grant, Lieutenant-General"
The note was carried through the Confederate lines and Lee promptly responded:
"April 7th, 1865. General: I have received your note of this date. Though not entertaining the opinion you express of the hopelessness of further resistance on the part of the Army of Northern Virginia, I reciprocate your desire to avoid useless effusion of blood, and therefore, before considering your proposition, ask the terms you will offer on condition of its surrender. R.E. Lee, General."
The correspondence continued up until that day the two generals met at the courthouse in Appomattox.
Lee had been facing the problem of fighting only a defensive strategy since the defeat at Gettysburg. Even though the previous year had cost the Union dearly at Petersburg/Cold Harbor/The Crater, and elsewhere, Lee was aware that Sherman had made it to the sea and was heading North. The fall of Richmond was the last straw.
There was no longer hope for victory. Without that hope, he was not going to ask his men to die in vain. He got the best terms that he could manage so that most of them could go home and try to rebuild after the war.
You ask: why not keep fighting?
With hope gone, what they were fighting for was not achievable.
With their bases of supply destroyed or captured, they were approaching the problem of having nothing to fight with.