The broken practice record
Of course, there's no substitute for practice. When you're talking about coping with stress, one way to improve your skills under fire is to reduce the amount of fire you're dealing with. This generally means that you can improve your ability to shrug off stressful situations by becoming familiar with them.
Think back to the first time you were ganked in a lane. It was probably so surprising that you froze and/or reacted very poorly. Even if you read articles on ganking and understood what could happen, it's pretty unlikely that your reactions were slow, your judgment was poor, and you died before you really understood what was going on. The intent of a gank is to bring a ganked player outside of his comfort zone, but there's a marked difference between someone who hasn't experienced a gank and someone who has dealt with ganks many times.
A friend was watching me play some time ago and he commented on my ability to use Flash on instinct. If you train up your ability to use panic abilities like Flash, Heal, or Barrier, you'll be much better-equipped to use them when you need them. I'm still very bad at using Zhonya's Hourglass, for instance, but I've been improving with practice.
Experiencing a wide variety of situations and practicing your reactions in those situations helps cool the fire that might otherwise cause panic. You can't prepare for everything, but practice helps you prepare for a wider range of circumstances and reduces the number of things that can take you by surprise.
Stop thinking so hard
Some people are just naturally good at letting their instincts take over. That's the key to really performing under stress. Of course, their instincts have to be good first (see practice, above). Even if they are thrown into unfamiliar situations, they just play as best they can, think about the situation when they have time, and apply that knowledge to their instinctual play, sometimes right in the middle of a match.
I've found that for a thinking guy like me, the best thing to do is to "pre-calculate" what can happen when I have time to think, but then play as much as I can in a sort of "autopilot" mode. Instead of thinking about the whole match or even most of what I'm doing, I focus most of my conscious mind on whatever it is that I'm trying to practice and let my skills do the rest of the work.
I'm aware that telling someone not to think is not really going to work. For thinking-type people, it's something we try to do as much as possible. Instead, we need to distract our conscious brains to allow our instincts to do more of the heavy lifting.
First, focus on one thing in the match. If there's some specific thing in the current matchup that requires reaction (such as skillshots you need to dodge), then focus on that. You can also focus on some specific thing you're trying to practice. If you notice that you're making mistakes on things that you're not focusing directly on, you probably need to work on whatever that is. For instance, if you're having trouble last-hitting or aiming skillshots for harassment, play some matches later on where you're focused on those things.
Once your mechanics for a particular element of the game are solid, though, you can pretty much completely avoid thinking about that part of the game directly. This is what I meant at the very beginning when I said that I just rely on improving my overall skill level.
Sometimes stressful things occur in a match. The more we think, "I just need to pull X off," the worse we tend to get at doing X. When I was first learning how to use Kassadin, using Riftwalk well was incredibly hard to do under pressure. It's not a normal skill, so my regular toolkit didn't really work for learning to cast it at the right times with my mouse in the right spot. As a result, I made a lot of misplaced teleports, usually causing my death.
It's more important that you stop focusing on stressful things (unless you're practicing those things) and focus on either something you need to practice or nothing at all. It's hard advice, but dealing with those sort of things is not easy. It's easy to say, "Stop thinking about what's stressing you," but it's another thing altogether to actually do it.
If you have time during a match, try focusing on things like your breathing or heart rate. If you can control your breathing, your body will start to level out. It isn't a perfect solution, but it does help me a lot.
Finally, if you can cope with stress well in life, it does spill over into other things you do. If you really want to improve your League of Legends
game, it might be a good idea to take some stress management classes or at least read some things on stress management. Of course, that's just good advice in general if you're having trouble coping with stress.
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