I think it has something to do with the sacrifice of this flesh day in and day out.
In another instance, Jesus said this in the context of denying all others in this life: Luke 14:26-27 (KJV)
26 If any man come to me, and hate not his father, and mother, and wife, and children, and brethren, and sisters, yea, and his own life also, he cannot be my disciple.
27 And whosoever doth not bear his cross, and come after me, cannot be my disciple.
1 Corinthians 15:31 (KJV)
31 I protest by your rejoicing which I have in Christ Jesus our Lord, I die daily.
Romans 12:1 (KJV)
1 I beseech you therefore, brethren, by the mercies of God, that ye present your bodies a living sacrifice, holy, acceptable unto God, which is your reasonable service.
Interesting side note, the gospels of Matthew and Mark don't use the word "daily".