"Estar a la que salta" means something like "being jumpy", but not only that. Somebody who "está a la que salta" probably is on the edge and nervous, and exhibits an irascible behavior, but that behavior is the real meaning of the expression and it can be used in other context without the person being upset or nervous.
Imagine that I'm really nervous due to an important exam or job interview and anytime that someone says something to me I yell back to them in frustration or I overreact. They could say
Diego está a la que salta. A nada que le dices responde a gritos Diego is so jumpy (or on edge). If you try to talk to him he'll yell back at you!
Another context: I'm frustrated, or upset, for whichever reason and I pay that on the people close to me. I reply angrily and bitterly to any of other people's comments. I criticize everything other people do, just because I'm upset and frustrated, not missing any chance to do so. Any time an opportunity of doing so comes around (think of it as a bunny that jumps or "salta" out of its rabbit hole) I'm going to shoot, I'm going to go for it. It would almost seem that I'm paying attention for these opportunities to show up to criticize, yell at others and discharge my frustration on them.
Another context in which you could use it: I'm with a group of friends and almost every time somebody says something I'm able to get a double meaning of it, making some kind of sexual joke. They could also say "estás a la que salta" meaning "You don't miss an opportunity to make a joke" or "It would seem that you are paying attention to this chances to show up to make another joke. You don't miss any of them". I guess that in English they wouldn't say I'm being jumpy, but maybe too cocky or annoying.
No se puede decir nada. Estás a la que salta y le encuentras el doble sentido a todo We can't talk! You don't miss a chance to find the double meaning in everything we say.
If you are in a meeting at work and somebody criticizes all your ideas, maybe he is not upset or nervous, maybe he is just a jerk or an adversary, but he is "a la que salta", eager to find a flaw in your ideas anytime you speak to make you look bad in front of the boss.
Paco estaba a la que salta en la reunión con cada una de mis ideas. Quería hacerme quedar mal delante del jefe porque competimos por un ascenso al mismo puesto. Paco didn't miss an opportunity to make each of my ideas look bad in the meeting. He wanted me to look bad in front of the boss because we are competing for a promotion.
It is not that the expression has two different meanings (like "never miss an opportunity or to be very touchy/sensitive" vs. "being nervous or annoyed"). It is as if I'm in a state in which I am nervous, annoyed, frustrated, etc. this leads me to behave in a certain way (jumpy, overcritical, overreacting, etc.) and not missing any opportunities to do so to others. It doesn't really mean "being nervous or annoyed" it means exhibiting this kind of behavior, "never miss an opportunity", which can be used in a different context.