Our spirit is the core of who we are. It is the center of our volition and our emotions. We also know that God is a spiritual being. To worship in spirit, then, is to do something that is beyond the physical. We do not worship by simply bowing our knees; we worship through a heart posture (Psalm 51:17). And our worship is in line with the worship going on in heaven (Psalm 148:1-2; Ephesians 6:12; Revelation 4:8).
Worshipping in truth means that we worship based on truth (see Romans 10:2). This includes the truth about who God is and what He does, hence why He is worthy of worship. It also includes the truth about our circumstances. We worship God even when we are experiencing heartache. When we worship Him, we do not forget about our hurt, but we worship even in the truth of our hurt. We also worship in joyful circumstances. Truth itself can be a means of worship; we worship God when we declare His truth.