This is a really important theme that you should be aware of right from the start. Chord substitutions make jazz sound like jazz, and they also make jazz playable.
First, chord subs to make life easier. Consider this:
How long does it take to learn how to improvise over those 15 chords? Answer: not nearly as long as you think, because it's actually only 3 chords. The first important area of chord subs is simplification. When improvising, just think Cm7 for all the chords in the first row, C7 for the second row and Cmaj7 for the 3rd row. That gives you the four important chord tones for improvising, and for the rest you use your scales, your imagination and your ears, rather than trying to approach each chord symbol differently.
For comping, you can put the same relationships into reverse. If you see Dm7-G7-Cmaj7 in the chart, feel free to use any of the extensions and alterations shown above, and more, in order to make a cool and interesting accompaniment. That's what I meant about making jazz sound jazzy. Just keep your ears open when playing with others - if you play a #5 while the soloist is playing a natural 5th, you'll hear what I mean!
But there will also be times when you want to simplify when comping, because the chart shows some 7b13 chord you can't grab in a hurry, or because you want to leave the soloist plenty of space. For this it's worth learning these chord grips:
Here you have minimalistic chord grips for the 4 chord types, on the top row with the root on the 6th string, on the bottom row with the root on the 5th - so with these two sets between the 3rd and 9th frets, you can play in any key. They all use only the root and the middle two strings, so it's a laid back, subtle sound. Mute the unused strings with your fretting hand. This lets you state the basics of the harmony through any tune. I find it lets me comp with confidence, knowing I always have these shapes to fall back on.
So what's happening here - why is it OK to leave out the extensions and alterations that the chart shows, and to add them when they're not in the chart? Ultimately, it's about jazz being improvised music. Unlike pop, rock and folk guitar, for a jazz tune there is no single "right" chord progression. As a jazz musician, you play it your way.
That being the case, it's fair to ask why jazz charts show extended and altered chord names at all - why not just show the basic 7th chords throughout if the player is going to be adding his own extensions and alterations anyway? There are three reasons why the full chord names get into the sheet:
Another important one is C7#5 = Gb7#5, called the tritone sub because the interval between the two roots is a tritone. Try it out in a ii-V-I in C: play Dm7-G7-Cmaj7, then compare the sound of Dm7-Db7-Cmaj7. You'll hear the cool altered 7th sound, plus the satisfying sound of the bassline going down in chromatic steps. It's also very easy to find on the fretboard.
You can use these substitutions when soloing too - but don't be in too much of a hurry. Make sure you can improvise from an Am7 scale over Am7 before you try to use it over other chords!