Why does my toilet make a high pitched whine when it refills?
Depending on the age of the toilet mechanism, you may have an easy fix. If it is a partially clogged valve (as suggested in the comment by ratchet freak) and if it is one of the newer type units, the upper part of the valve can be easily accessed.

Many toilet filler valves look something like this one



The top cap comes off these valves, usually by pressing down slightly with your palm and rotating counterclockwise. Immediately below the cap is usually a strainer filter that can get clogged. Cleaning this out usually solves the problem. Then just reinstall the cap.

If you have a different type of valve, you need to ascertain whether that is where the noise is coming from. With the lid off the tank, you may be able to confirm the source of the sound. If it is the valve, you may be able to determine what part of the mechanism is causing the obstruction.

If the cause is the valve, but the cure is not readily apparent, it probably is worth changing the mechanism, which is cheap and fairly easily done. The repair kits have complete instructions and basically involve


turning off the water supply
draining the tank
uncoupling the supply line from the bottom of the mechanism
unscrewing the nut that holds the mechanism on (from the outside bottom of the tank)
inserting the new mechanism in the hole and attaching the nut
reattaching the supply line
turning on the water supply
adjusting the max fill level


Not too hard for even an amateur DIYer. Just be ready to quickly turn off the shutoff valve if there are any leaks (even a drop), and patiently redo any joints that are not absolutely watertight (the instructions will explain about teflon tape, etc.)