How can I remove the bottom bracket from this old road bike?
This type of bearing is not serviceable. The plastic piece is the seal that retains the grease and keeps water and dirt out. The crank needs to be pressed out to remove the bearings. Try to tap the crank with a soft faced hammer. the bearing on the opposite side should be pushed out by the crank. Installing the new bearings will require some care. I have had good success using standard and deep sockets to drive in the new bearings.