The single most important attribute that a pitcher MUST have to be successful is confidence. You can have immense talent but you wont be successful if you lack confidence. Conversely, you can have marginal talent and still be successful if you are supremely confident. It wont always happen, but at least its possible.
The way to develop confidence is thru success. You have to believe in your training process and buy into what you are being taught. Obviously this requires the right teacher......and that is where the process breaks down for so many young pitchers. There is just so much bad information out there as it relates to pitching instruction. I describe pitching as both an art and a science. Envision the process of pitching as a pie that has been cut into 8 pieces. Its not hard to find a pitching instructor that knows about 1 or 2 pieces of the pie and can teach about that. But its really hard to find someone that can teach about the whole pie. But I digress..... Assuming that you have located a good instructor, and assuming the young pitcher has bought into the training process that is being presented to him, the next step is preparation. You prepare by doing drill work to solidify your mechanics. You build arm strength with a throwing program. You build your body with strength and conditioning work. You throw bullpens to establish command of the fastball and to learn how to throw effective off-speed pitches. You get comfortable with different grips. Once those things have been accomplished you are ready for game situations.......and you have the tools to be successful.
One thing that every pitcher has to learn is what to focus on. If you are doing bullpen work you focus can be one of two places - either on what is happening on the mound OR what is happening at the plate. But not both. When you get into a game the focus HAS to be on what is happening at the plate - both in terms of pitch location and movement (or lack thereof). Assuming that the focus is in the right place, the next thing is that EVERY pitch must be thrown with conviction. If a pitcher is not fully convinced that he is throwing the right pitch the chances are it wont work out well. This can get a little tricky if the pitcher doesn't have the freedom to shake off a pitch. If a coach is calling pitches its very important that there be a running dialogue during the game between the pitcher and the coach calling pitches to stay current about what is working and what isn't. Poor communication can cost you a game if the wrong pitch is called (and thrown) at a critical point. I tell all the pitchers I work with that they need to OWN every pitch they throw. They need to understand their own strengths (and weaknesses) and know how they are most likely to get hitters out. You would be amazed how many HS and college pitchers don't know what their most effective pitch is. Awareness is a critical thing in a game on the line situation.
Assuming that a pitcher has successfully navigated thru all the water described above, it now gets down to execution. I'm a big believer in positive imagery. You have to see yourself (in your mind) throwing the chosen pitch and getting a positive outcome BEFORE you actually do it. Next, you have to be in the proper mindset - and this may vary depending on the emotional makeup of the individual. For me, the bigger the situation the more I would downplay it in my mind. I would tell myself that I was pitching in a meaningless intrasquad game in order to stay relaxed. Assuming I agreed with the pitch call (I could shake off anything I didn't like) I would envision myself throwing the pitch exactly how I wanted to, then take a deep breath, deliver the pitch, and react accordingly. Sometimes that meant backing up 3B but most times it meant seeing something good happen.
That was the routine that worked for me. It could be way different for someone else.