Why the MPU6050 gyroscope read write register value is 0x08 for a full scale of 500 degrees per second?
The i2c address is 0x68 or 0x69. Some modules with the MPU6050 have indeed 0x69, but most have 0x68.

When searching the datasheet for the word "sleep", this sentence is easy to find: "Note: The device will come up in sleep mode upon power-up".

When the MPU6050 powers up, the SLEEP bit is 1 and the sensor is in sleep mode. Therefor writing a 0 is to get it out of sleep mode. After that you can set the sensor as you want.

The FS_SEL[1:0] is a notation that is used a lot in datasheets. It is an indication that only two bits are used. It is not the bit-position in the register. Those bits are located at Bit3 and Bit4. To turn on the lower bit of FS_SEL, Bit3 has to be set to 1, thus a value of 0x08 is needed for the register.