Toilet flushes ok, but then to fill tank, shut off valve has to be turned off and then back on. Why?
Your toilet fill valve has failed. You probably will need to replace it.
Whether it can be cleaned and rinsed out or adjusted or repaired, or must be replaced, depends on the type of fill valve. There are over a dozen common styes. Do a net search for "toilet fill valve images" and you will see what I mean.
You will also find the picture that matches your valve, which will lead you to the manufacturer, which will probably lead you to a tutorial on what you can do to maintain it. I can't tell you exactly what to do unless you post a picture of your fill valve.
The reason I think you will probably need to replace it is, locking up until the main water pressure is removed is a strange failure mode for a fill valve. However if your fill valve is a type that can be cleaned or adjusted, try that first (if you can find out how to take it apart).
Don't forget to close the supply valve before working on it, and if you have to temporarily open the supply valve to flush the line or whatever, hold a heavy towel or rag over the partially disassembled fill valve to avoid spraying the room.