Always seemed kind of obvious what Jesus meant. Jesus also says that that no one can serve two masters, He also says what does it profit someone to gain the whole world but forfeit their soul, and also that whoever keeps his life shall lose it, but whoever gives it away shall find it.
To deny oneself is straight forward. It means to deny ourselves, that we don't come first. That our calling as the people of Jesus Christ is to lay down our lives in love, sacrifice, and serve toward others. It's what St. Paul says, "regard others more highly than yourselves". Or what the Lord Himself says in the Sermon, that we should turn the other cheek, repay no one evil for evil, to bless those who curse us, to pray for those who persecute us, to love our enemies, to give without asking anything in return. To be merciful even as our Father is merciful, for He is kind to the thankless and the wicked, sending His rain upon the just and the unjust alike.
Fasting can be part of this, fasting is part of the Christian walk after all. But it's not the be-all, end-all of the cross-shaped life Jesus calls us to live here in the world. After all pious acts without love are meaningless. If I fast, but ignore the "weightier matters" such as mercy and justice, then I am nothing but a hypocrite. That wouldn't be denying myself, that would be glorifying myself and thinking myself righteous--as though abstaining from food and drink could bring me closer to God and His righteousness. Fasting is a good and ancient practice of the Church, but it's just one practice among many, and the heart of the Coram Hominibus (how we live out our Christianity into the world) is the Great Commandment: That we love our neighbors as ourselves.
To echo St. Paul: If I fast, but have not love, then I have nothing.
-CryptoLutheran