What does the Bible mean by 'dying to self'?

The concept of "dying to self" is found throughout the New Testament.
It expresses the true essence of the Christian life, in which we take
up our cross and follow Christ. Dying to self is part of being born
again; the old self dies and the new self comes to life (John 3:3–7).
Not only are Christians born again when we come to salvation, but we
also continue dying to self as part of the process of sanctification.
As such, dying to self is both a one-time event and a lifelong
process.

Jesus spoke repeatedly to His disciples about taking up their cross
(an instrument of death) and following Him. He made it clear that if
any would follow Him, they must deny themselves, which means giving up
their lives—spiritually, symbolically, and even physically, if
necessary. This was a prerequisite for being a follower of Christ, who
proclaimed that trying to save our earthly lives would result in our
losing our lives in the kingdom. But those who would give up their
lives for His sake would find eternal life (Matthew 16:24–25; Mark
8:34–35). Indeed, Jesus even went so far as to say that those who are
unwilling to sacrifice their lives for Him cannot be His disciples
(Luke 14:27).

The rite of baptism expresses the commitment of the believer to die to
the old, sinful way of life (Romans 6:4–8) and be reborn to a new life
in Christ. In Christian baptism, the action of being immersed in the
water symbolizes dying and being buried with Christ. The action of
coming out of the water pictures Christ's resurrection. Baptism
identifies us with Christ in His death and resurrection, portraying
symbolically the whole life of the Christian as a dying to self and
living for and in Him who died for us (Galatians 2:20).

Paul explains to the Galatians the process of dying to self as one in
which he has been "crucified with Christ," and now Paul no longer
lives, but Christ lives in him. Paul's old life, with its propensity
to sin and to follow the ways of the world, is dead, and the new Paul
is the dwelling place of Christ who lives in and through him. This
does not mean that when we "die to self" we become inactive or
insensible, nor do we feel ourselves to be dead. Rather, dying to self
means that the things of the old life are put to death, most
especially the sinful ways and lifestyles we once engaged in. "Those
who belong to Christ Jesus have crucified the sinful nature with its
passions and desires" (Galatians 5:24). Where we once pursued selfish
pleasures, we now pursue, with equal passion, that which pleases God.

Dying to self is never portrayed in Scripture as something optional in
the Christian life. It is the reality of the new birth; no one can
come to Christ unless he is willing to see his old life crucified with
Christ and begin to live anew in obedience to Him. Jesus describes
lukewarm followers who try to live partly in the old life and partly
in the new as those whom He will spit out (Revelation 3:15–16). That
lukewarm condition characterized the church of Laodicea as well as
many churches today. Being "lukewarm" is a symptom of unwillingness to
die to self and live for Christ. Death to self is not an option for
Christians; it is a choice that leads to eternal life.