How to Wire New 1 Way Switch From 2 Existing 3 Way Switches
I am afraid that the way the photos are taken I can't clearly see what wires go where.  But I have a fairly safe guess for your answer.  

When you have two three way switches on a light, you have a circuit like this:  



Note that the common terminal on the switch on the left is connected to the constant-hot power source (let's call it the line side), and the common on the switch on the right is connected to the light or whatever, let's call it the load side.  

(The actual physical layout, wire routing, and the color coding may be quite a bit different than the picture - the illustration is just to show how the circuit is put together.)  

It sounds like you tapped the wire going to the common terminal on the middle switch to connect to the new switch on the right, and switch happens to be on the load side of the 3-way switch circuit.  

If that switch happened to be on the line side of the 3-way switch circuit, and you tapped the wire on that common terminal for your new switch, you'd be getting constant hot at the third switch, which is what you want;  that new switch / dimmer alone would control the new light.  

But as it is, you're supplying that right-side dimmer switch with whatever is coming through to the load side of the middle 3-way circuit - no power, full power, or dimmed power, depending on the position of the switches. 

There is of course one hope for an easy solution.  If the three-way switch on the left happens to be on the  line side of that three way circuit, you can tap the common from that switch instead, and you're set.  

If you don't have a constant hot in the box - just travelers for the three way circuits entering the box - you have a bit of a challenge.  You will have to either run another cable to supply power, or you'll have to convert one (or both) of the three way circuits to some kind of wireless switching, freeing up the travelers.  Then you can use one of those wires for constant power.