The diagram below shows the signal from one of the PWM pins on the Arduino.
If we specify a value in the analogWrite that is somewhere in between 0 and 255 then we will produce a pulse. If the output pulse is only high for 5% of the time then whatever we are driving will only receive 5% of full power.
If however the output is at 5V for 90% of the time then the load will get 90% of the power delivered to it. We cannot see the LEDs turning on and off at that speed, so to us, it just looks like the brightness is changing.