The lesson works through John 15, where Jesus calls Himself the true vine and His Father the vinedresser. Every branch that bears fruit is pruned to bear more, and branches that do not abide are thrown away. Apart from Christ we can do nothing; those who abide in Him and whose lives are shaped by His words bear much fruit, and so the Father is glorified. Abiding in His love means keeping His commandments, especially the command to love one another as He has loved us.
The slides explain abiding as a picture of believing, trusting, receiving and submitting every part of life and thought to Christ and His lordship. It also means enjoying Him, fellowshipping with Him and resting in Him. John 6 is linked in: whoever feeds on Christ abides in Him. To abide in Christ is to live and die for Him, since believers no longer live for themselves. Christ chose and appointed His people to go and bear fruit that remains.