The article "Who
ReallyHaveaHeavenlyCalling?" Watchtower 91 3/15, pages 19-22 have a description of how the anointing process happens. Here are some parts of that article:
The 144,000 "have an anointing from the holy one," Jehovah. (1 John 2:20) It is an anointing to a heavenly hope. God has ‘put his seal upon them and has given them the token of what is to come, that is, the spirit, in their hearts.’—2 Corinthians 1:21, 22.
Yes, those with the heavenly calling have the testimony of God’s spirit to that effect. Concerning this, Paul wrote at Romans 8:15-17: "You did not receive a spirit of slavery causing fear again, but you received a spirit of adoption as sons, by which spirit we cry out: ‘Abba, Father!’ The spirit itself bears witness with our spirit that we are God’s children. If, then, we are children, we are also heirs: heirs indeed of God, but joint heirs with Christ, provided we suffer together that we may also be glorified together." It is by means of God’s spirit, or active force, that anointed ones cry out, "Abba, Father!"
The chief evidence that a person has been anointed to the heavenly calling is a spirit, or dominant sense, of sonship. (Galatians 4:6, 7) Such an individual is absolutely sure that he has been begotten by God to spiritual sonship as one of the 144,000 joint heirs of the heavenly Kingdom. He can testify that his heavenly hope is not his own cultivated desire or his imagination; rather, it is from Jehovah as a result of the action of God’s spirit toward him.—1 Peter 1:3, 4.
Under the influence of God’s holy spirit, the spirit, or dominant attitude, of anointed ones acts as an impelling force. It moves them to respond positively to what God’s Word says about the heavenly hope. They also respond in a positive way to Jehovah’s dealings with them by means of the holy spirit. Thus, they are sure that they are God’s spiritual children and heirs.
When anointed ones read what God’s Word says about his spiritual children and the heavenly hope, their spontaneous inclination is to say within themselves, ‘This means me!’ Yes, they respond joyfully when their Father’s Word promises a heavenly reward. They say, ‘That means me!’ when they read: "Beloved ones, now we are children of God." (1 John 3:2) And when anointed ones read that God has brought people forth "to be certain firstfruits of his creatures," their mental inclination is to respond, ‘Yes, he brought me forth for that purpose.’ (James 1:18) They know that they have been "baptized into Christ Jesus" and into his death. (Romans 6:3) So they have the firm conviction of being part of Christ’s spiritual body and entertain the hope of undergoing a death like his and being resurrected to heavenly life.
Futher down in the article it says:
If Jehovah has cultivated in you the undeniable hope of heavenly life, you are counting on that. You offer prayers in expression of that hope. It engrosses you, and you cannot get it out of your system. You have consuming spiritual aspirations.