What a good question ... makes me think too. Well, I believe that there is some physical element to dying daily (which is not spiritual). Have you ever met anyone who is successful and who have not had any major failures in their life? It's difficult to know what humbling and dying daily is when you are always up there on a pedestal looking down. Most great people in the bible had to die to themselves, their achievements and their abilities because of circumstances. Joseph had his multi-colored coat and he was his father's favorite. Circumstances happened, his brothers betrayed him and sold him as a slave and things appeared to go downhill then. But his character was being built because of all the choices that he had to make daily. He had to die daily to himself because of "physical" things that were happening around him. But he had a choice, to degenerate and just live life as if there is no God or to continue to hope and make the right choices in spite of the circumstances.
Moses had to die to his abilities when he tried to save his people with his own strength. He had to flee and humble himself until he became a man who does not believe in himself anymore before God could use him. He died to himself because of circumstances. But he made the right choices. Instead of running away from God, he ran to the mountain of God (mount Horeb) [Exodus 3:1].
I had to die to myself too. I've always been sheltered from many things when I was young. My parents were good to me, circumstances were good to me and God was good to me. But God allowed me to understand the limit of my abilities and I dropped from my pedestal and I saw God's glory. I learned to die to myself and to live unto Him.
The spiritual aspect of dying daily involves our spirit man. We have to become close to God. When we were born again, God made everything new and gave us a new spirit. But our mind was not renewed and our spirit was not being fed. When we feed ourselves with the Word of God and learn to be led not by our flesh but by our spirit, we learn to die to self.
13 For if ye live after the flesh, ye shall die: but if ye through the Spirit do mortify the deeds of the body, ye shall live.
14 For as many as are led by the Spirit of God, they are the sons of God.
This involves letting our spirit grow and aligning our will to the Will of God. Jesus died daily because He said in [John 5:30]
"I can of mine own self do nothing: as I hear, I judge: and my judgment is just; because I seek not mine own will, but the will of the Father which hath sent me."
Dying daily involves learning God's will and seeking not our own will but His will because we know that His will is the best and because we love Him.
Dying daily involves making a conscientious effort to be obedient to His Word and to His Will and to walk in His ways.
[2 Corinthians 10:3-6]
3 For though we walk in the flesh, we do not war after the flesh:
4 (For the weapons of our warfare are not carnal, but mighty through God to the pulling down of strong holds;)
5 Casting down imaginations, and every high thing that exalteth itself against the knowledge of God, and bringing into captivity every thought to the obedience of Christ;
6 And having in a readiness to revenge all disobedience, when your obedience is fulfilled.
Sin and disobedience usually begin with our thoughts, and with our imaginations. That was how the devil approached Adam and Eve and that is still how he operates now. The strongholds mentioned in the above scripture is not demonic powers of the air but our minds .. imaginations and thoughts that exalt itself against the knowledge of God, that goes against His Word. When we have that coming to us, we bring those thoughts into captivity to the obedience of Christ to the obedience of His Word. When we do that, we learn to die daily to ourselves and to live for Him.