The soft pedal on a grand piano makes the notes played "softer", that is, quieter, with not such a sharp attack.
The way the pedal works is very simple, but before I can explain it I have to explain something about the strings in a piano. For the bass notes, the lowest notes on the piano, there is one string for each note. For the middle notes, there are two strings for each note, tuned identically. When you strike one of the middle notes two strings sound. For the upper notes, there are three strings per note, all three tuned identically for each note. The reason for this is to even out the volume levels so that the treble notes and middle notes and bass notes will all sound at about the same volume when struck with the same force.
So back to the soft pedal. The soft pedal on a grand piano works by physically shifting the entire action, that is, the entire keyboard, and all those crazy felt hammers and
everything sideways just a little bit. With the keyboard
and hammers shifted over, when a note is played in the middle of the keyboard, only one of the two strings is struck. In the upper register, only two of the three strings are struck. And in the bass, the hammers don't hit the strings dead-on, but glance off the side just a little bit. The effect of all this is to make the piano sound not just quieter, but also a little bit different than it normally does.
The soft pedal on some upright pianos does not seem to work the same way.