There is a known medical condition known as hematidrosis in which a person has blood or blood pigments present in his sweat. This happens when ruptures in the skin occur that allow blood or blood pigment to move from the capillaries into the sweat glands.
On the literal level, it would seem that Jesus, suffering the extreme stress of the Agony in the Garden, began to suffer from hematidrosis. On the spiritual level, this event foreshadows the shedding of his blood for us on the cross.
On the apologetic level, the recording of this detail reveals Luke as the careful investigator and historian that he was. Hematidrosis is rare enough that–given the state of medical knowledge at the time–Luke (who was a doctor) may not have known about it, and so he recorded the detail. Though even if he did know about the condition, it still reveals his careful reporting of detail.