Bernt beat me to it... he's right. It's a very low-tech, non-electrical fiber optic backup system to let you know your bulbs are working. I'll add that the system is very likely explained in the owner's manual; definitely in the service manual.
Bulb-wise... Sounds like you have a short somewhere. I've learned that shorts or burnouts anywhere in that circuit can cause weird gremlin behavior. Recently my turn signal stopped blinking, turn bulb just lit solid. Found and replaced an unrelated burned out bulb, I think it was a running light bulb... then turn bulb blinked again.
So step one with any such lights issues, for me anyway, is checking all the turn/brake/running light bulbs everywhere. All of them. Replace any burned out bulbs.
If they all work: those bulb sockets get dirty/corroded and also very hot. The sockets often melt and warp. Either of those could be your problem. First, remove the bulbs and check for dirt/corrosion or melting. If not melted, then clean the contacts inside with Electrical Contact Cleaner and Q Tips. In a pinch, use alcohol. If really dirty, use an old toothbrush. (To go the extra mile, put Dielectric Grease on the socket and bulb contacts.) Then test.
If melted/warped, replace the sockets. If you still have the problem, personally I'd start looking at the electrical diagram and try to figure it out. You might have a bad headlight switch or brake light switch, or loose wires somewhere in the circuits.
Bulb-wise... Sounds like you have a short somewhere. I've learned that shorts or burnouts anywhere in that circuit can cause weird gremlin behavior. Recently my turn signal stopped blinking, turn bulb just lit solid. Found and replaced an unrelated burned out bulb, I think it was a running light bulb... then turn bulb blinked again.
So step one with any such lights issues, for me anyway, is checking all the turn/brake/running light bulbs everywhere. All of them. Replace any burned out bulbs.
If they all work: those bulb sockets get dirty/corroded and also very hot. The sockets often melt and warp. Either of those could be your problem. First, remove the bulbs and check for dirt/corrosion or melting. If not melted, then clean the contacts inside with Electrical Contact Cleaner and Q Tips. In a pinch, use alcohol. If really dirty, use an old toothbrush. (To go the extra mile, put Dielectric Grease on the socket and bulb contacts.) Then test.
If melted/warped, replace the sockets. If you still have the problem, personally I'd start looking at the electrical diagram and try to figure it out. You might have a bad headlight switch or brake light switch, or loose wires somewhere in the circuits.