Why did Gen. Lee (Civil War) surrender to the U.S?

All these answer are right to some extent but there was one critical moment when IIRC General Preston, his artillery officer tried to convince him to take to guerrilla warfare.

But Lee was already thinking far ahead. He, like Grant and Lincoln knew that the Union had to be healed back into a functioning whole. The best interest of the South and North now rested with the rapid reintegration of South back into the Union.

Arguably, it was in the four years he lived after the war in which he became the hero that the Myth of the Lost Cause had tried to foist on him. He used his enormous moral authority and charisma to stamp out any violence and speed reintegration.

If he'd lived another 10 or 15 years, the history of American might have been much better. Instead, in 1877, we got Civil War 1.5, the Democrats Strike back in which they murdered or drove out all the Republicans from the South while at the same launching a similar war of terror in the North to establish the all-white Unions and lock in the big city Democrat machines. The mostly Irish thugs in the North that murdered "scabs" and fought the Pinkerton's who defended them, became the basis of America organized crime, which formed a complex with the unions and the party machines.

It's possible Lee could have prevented some of that had he lived longer. He did try to ensure that Freedmen were allowed their 2nd Amendment rights as well as opposing the early Jim Crow laws that sought to prevent skilled Freedman like blacksmiths from competing on an equal basis with whites.

Generals will order their soldiers to "fight to the death" only when there is a military or political advantage to be gained. This was not the case here.

At the Alamo, some 174 Texans defended a fort to the death and inflicted several hundred casualties (a multiple of their own number) on the Mexicans. This weakened Santa Anna to the point where he could not pursue the main Texas force under Sam Houston until he received reinforcements. Houston also received reinforcements and won the battle of San Jacinto. A similar story could be told for say, the "fight to the death" defense of Stalingrad, to buy time for the encircling forces.

In the case of the Confederates, they had just been ousted from their fortified positions around Richmond, which was the Confederate capital. Half starved soldiers were fleeing west on foot, many without shoes. There were only about 30,000 left from an original strength of over 60,000. (And the decline in Confederate strength was more than 50% because many of these men were replacements for fallen veterans, supported by limited ammunition and almost no artillery.)

Their objective was Lynchburg, which supposedly offered them food and transportation to the mountains where there was at least a hope of a guerrilla resistance. Neither was present there, and the Confederates were caught in open country, outnumbered four to one, (not the earlier two to tone, in fortified positions), with hordes of Union cavalry ready to cut down any stragglers or "escapees" from the main fight. Under the circumstances, a "fight to the death" would have been suicide, without the chance of winning or even inflicting meaningful casualties.

In 1864, with his line broken at Spotsylvania, the same General Lee asked his men to "fight to the death" to close the breach at a place called the "Bloody Angle" by offering to personally lead the counterattack, to allow his engineers to build a new defense line. Their success prolonged the war almost a year. No such advantage offered itself around Lynchburg.

Robert E. Lee

to

Jefferson Davis

Richmond, Virginia April 20, 1865

Mr. President... ...At the commencement of the withdrawal of the army from the lines on the night of the 2d, it began to disintegrate, and straggling from the ranks increased up to the surrender on the 9th. On that day, as previously reported, there were only seven thousand eight hundred and ninety-two (7892) effective infantry. During the night, when the surrender became known, more than ten thousand men came in, as reported to me by the Chief Commissary of the Army. During the succeeding days stragglers continued to give themselves up, so that on the 12th April, according to the rolls of those paroled, twenty-six thousand and eighteen (26,018) officers and men had surrendered. Men who had left the ranks on the march, and crossed James River, returned and gave themselves up, and many have since come to Richmond and surrendered. I have given these details that Your Excellency might know the state of feeling which existed in the army, and judge of that in the country. From what I have seen and learned, I believe an army cannot be organized or supported in Virginia, and as far as I know the condition of affairs, the country east of the Mississippi is morally and physically unable to maintain the contest unaided with any hope of ultimate success. A partisan war may be continued, and hostilities protracted, causing individual suffering and the devastation of the country, but I see no prospect by that means of achieving a separate independence. It is for Your Excellency to decide, should you agree with me in opinion, what is proper to be done. To save useless effusion of blood, I would recommend measures be taken for suspension of hostilities and the restoration of peace.

I am with great respect, yr obdt svt

R. E. Lee Genl

As you can see, most men were not willing to fight. The war was hopelessly lost, and continue to fight would be "useless effusion of blood" (unlike Alamo. other Confederate armies could get no reinforcement from devastated country).

Actually, only fanatic Jefferson Davis prevented restoration of peace two month earlier, after Hampton road conference, which could save South humiliating military capitulation and loss of thousands lives.

UPDATE

I wanted to correct TechZen answer, but my score is not high enough to comment; so I am doing it in my answer.

Artillery officer who was trying to convince Lee to take to guerrilla warfare was not General Preston; his name was brigadier General Edward Porter Alexander. This is how Lee responded to him:

“ If I took your advice, the men would be without rations and under no control of officers. They would be compelled to rob and steal in order to live. They would become mere bands of marauders, and the enemy’s cavalry would pursue them and overrun many sections they may never have occasion to visit. We would bring on a state of affairs it would take the country years to recover from. And, as for myself, you young fellows might go bushwhacking, but the only dignified course for me would be to go to General Grant and surrender myself and take the consequences of my acts.”

Robert Lee was the great general. But his greatest service for his country was not what he did on the battlefields as army commander, but what he did not allowed to do. It was this surrender, which prevented guerrilla warfare. This was his service for both South and North. Jefferson Davis did not take his advice, and insisted on war continuation. But fortunately, confederate generals did not obey his orders, they followed Lee.

To augment KorvinStarmast's excellent answer, it's worth looking at a map to understand just how over the war was.

Source Perry-Castañeda Library Map Collection

Since the beginning of the war, the South had been blockaded by sea cutting off cotton exports and arms imports. This left them with increasing financial problems, and fewer and fewer quality arms to fight with.

In 1863, with the Union victory at Vicksburg, the Confederacy was split in two cutting off the manpower and supplies from Texas and the Western Confederacy.

In 1864 the Confederacy was split again in Sherman's March To The Sea leaving the Carolinas and Virginia left to fight alone. In December, The Army Of The Tennessee was rendered ineffective as a fighting force in the Battle of Nashville.

1865 found Lee with the last effective Confederate army in the Eastern Confederacy. He was being chased by the Army of the Potomac and knew Sherman would be coming up from the south. In April he'd failed to raise the siege of Richmond (the Confederate capital) at the Battle of Five Forks and realized the capital was lost.

The mission of the Army Of Northern Virginia was to attack Washington D.C. (and thus win the war) and defend Richmond. It could no longer do the former, and it had failed at the latter.

Tactically, Lee knew he was totally outnumbered and outgunned with an almost certainty of being surrounded. His capital was lost, he had no hope of resupply. His enemy was lavishly supplied. There was no hope of winning the war.

Personally, to continue would only further harm his men and his beloved Virginia. Sherman's army was tearing through the Eastern Confederacy and he did not want to see Virginia suffer the same fate as Georgia. If he fought on, his men would die by starvation or by a bullet. If he surrendered, his men would be allowed to lay down their arms on good terms.

The officers to give their individual paroles not to take up arms against the Government of the United States until properly exchanged, and each company or regimental commander sign a like parole for the men of their commands. The arms, artillery and public property to be parked and stacked, and turned over to the officer appointed by me to receive them. This will not embrace the side-arms of the officers, nor their private horses or baggage. This done, each officer and man will be allowed to return to their homes, not to be disturbed by United States authority so long as they observe their paroles and the laws in force where they may reside.

They were even allowed to keep their horses and mules for farming, and were given rations for the trip home.

Lee's surrender was just the first. For over a month, holdout armies and governments of the Confederacy would fight and surrender.