Exegesis:

kai genomenos en agōnia ‘and becoming greatly distressed, or, anguished.’

agōnia ‘anxiety,’ ‘anguish,’ ‘distress.’

ektenesteron prosēucheto ‘he prayed more fervently.’

ektenōs ‘eagerly,’ ‘fervently.’

kai egeneto ho hidrōs autou hōsei thromboi haimatos ‘and his sweat became like drops of blood.’

hidrōs ‘sweat,’ ‘perspiration.’

thrombos ‘small amount of blood,’ ‘clot,’ here ‘drop.’

katabainontos epi tēn gēn lit. ‘going down to the ground,’ hence ‘falling on the ground,’ going with haimatos.

Translation:

Being in an agony, or, ‘in great anguish/distress,’ ‘his heart was oppressed’ (Uab Meto). The vehemence of the emotion may colour the rendering of the verb, cf. e.g. ‘broken with anguish’ (cf. Navajo). The phrase indicates cause, cf. ‘his agony made him pray’ (Sranan Tongo).

More earnestly, or, ‘even more intensely and seriously,’ i.e. than previously.

Falling down upon the ground, though grammatically going with ‘blood,’ may in translation often better be taken with ‘drops,’ and then be rendered, ‘trickled/dripped to the ground’ (Balinese).

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.