Most people who took piano lessons as a kid, including me, grew up learning to read music exactly as it is written on the printed page. Being able to read music is a valuable skill, and I am delighted I learned that skill at an early age.
But there is a downside to only being able to read music without understanding what you are reading. A couple real-life examples illustrate the point:
What happens when you are playing and your sheet music slips off the piano and onto the floor? Unless your name is Victor Borge and you can turn the situation into comedy, you will probably find the situation extremely embarrassing. I have seen it happen several times to pianists who play well but who don't know how to improvise, and it's not a pretty site. It happened to me once years ago as well. I was accompanying a singer and someone opened a side door, letting a gust of wind sweep into the auditorium and right across my piano. The sheet music scattered onto the stage and a couple pieces blew off the stage. If I hadn't understood the music and had a grasp of it's harmonic form, I would have had to stop, pick up the music, get it back in order, and so forth, delaying the soloist and the performance. As it turned out I had several people ask me how I kept playing without the music in front of me. I replied that I knew the chord progressions of the song, so was able to "wrap the chords around the singer" and therefore keep the song going.
So what are chord symbols, and how do they work?
Chord symbols are a shorthand way of writing what is going on harmonically in a song. For example, if I were to write the chord symbols of the first line of What Child Is This? (also known as Greensleeves and several other titles) in the key of Am, I would write:
Am G F E7 -- which corresponds to the first line of the song and would appear directly above the melody line in the treble clef, so all the pianist would need to do would be to read the melody (tune) of the song -- not all the supporting notes.
I think you can see that once you know a few chords this would be infinitely easier to remember than the entire score of the song. Not only that, but that chord progression -- A, G, F, E7 -- repeats several times during the song, so once you know the form of the song, you have a huge advantage over someone who is chained to the written music and has no idea about the logic of the song.
So how does a person learn this "musical shorthand"? It's no secret -- there are books galore on learning chords, plus web sites that teach chords, or you could even pick up a chart of chords in your local music store.
Then buy a "fake book" -- a songbook with hundreds or even thousands of songs, each song showing just the melody of the song with the chord symbols listed above it. Each song alone would be known as a "lead sheet", but cumulatively the collection of songs is known as a "fake book". Then every day play a dozen or so songs just with the melody in your right hand and the chord in your left hand. It will sound barren at first, but you're learning how it works.
After a couple weeks of that, instead of playing the melody in your right hand, sing the melody (doesn't matter at all how it sounds) and use your right hand to break up the notes of the chord you are playing in your left hand. Once you get the hang of it, you can start breaking up the chord in both hands and experimenting with various rhythm patterns.
I am not saying it's easy; I am saying it's fun and exciting and that it is worth it many times over!