Your question comes from Luke 22:44,   "And being in an agony he prayed more earnestly: and his sweat was as it were great drops of blood falling down to the ground."  Luke is the only gospel that gives this detail.  This verse is not found in the Vatican manuscript of the Bible. It was in the original Sinaitic manuscript, but was crossed out by a later critic. 

So, the remaining questions are:

1)  Could  the bloody sweat have happened?  Medical science says yes.  It is rare, but it has been reported.  Extreme sorrow and agony can cause the tiniest blood vessels near the surface of the skin to break. The blood can reach the outside of the skin, thus producing a bloody sweat. The drops, (called thromboiclots), are a symptom of a condition called diapedesis. This is caused by extreme tension of the nervous system.  It once afflicted Charles IX of France.

2)  Since Jesus willingly came to earth for the purpose of providing a ransom for mankind, he knew he would die.  Why was his agony in the Garden of Gethsemane so intense?   There were two reasons:

  • He knew he would be crucified as a blasphemer, which was difficult to bear.  Jesus accepted this, and told his Father, "thy will be done," Matthew 26:42. 
  • He also need reassurance that he had completed his sacrifice totally perfectly in the eyes of his Father. He had the weight of the whole world on his shoulders, as we say.  It was a heavy burden. And God did hear his prayers and comforted him. Hebrews 5:7 tells us, “During the days of Jesus’ life on earth, he offered up prayers and petitions with fervent cries and tears…and he was heard because of his reverent submission.” Therefore, with courage and composure, Jesus left the garden and completed his sacrifice unto death. 

What a wonderful Savior!  We are thankful for his sacrifice and for his example to follow.