Psychotherapy is, at it's core, designed to change people. We do this through helping people have new experiences that are more in line with their goals of who they want to be and how they want to operate in the world. Our brain is shaped largely by experience. If you have the experience of practicing piano every day then the pathways of neurons (brain cells) that are used to play piano get stronger. Think of neurons like muscles-- the more you work them out, the stronger they get. So if you work out the same "set" of neurons (a "neural pathway") every day, say by practicing piano, then those get stronger and stronger and easier to activate. This is how we build proficiency in things, like playing baseball or practicing piano, or even being good at making small talk.
Some people grow up in families where they don't have certain experiences like being able to talk about their feelings, or being able to ask for what they need from others. When those experiences are missing in childhood those neurons that are associated with that behavior are weak and hard to activate. Psychotherapy aims to provide experiences that were missing in childhood (or adulthood) that are needed to build adaptive behaviors that help us lead happy and fulfilling lives. So for example a person who grew up in a house where it was not OK to talk about one's feelings gets to talk openly about how they feel in therapy. That in turn exercises those neurons and strengthens that neural pathway so that talking about one's feelings becomes easier and easier.
In a very real sense psychotherapy is like hiring a personal trainer at your gym-- a person who can learn about how you would like to be (versus where you are now), set up an "exercise routine" to work out those muscles (neurons) and take you through those steps so that you can develop the muscles (skills) that you want. If we were to take a "before" and "after" picture of your brain we could actually see those neuronal changes that are a result of psychotherapy. As a matter of fact, studies have shown that one impact of psychotherapy is that the connections between the frontal lobe (which involves planning, organizing, regulating emotions, understanding consequences, controlling impulses and lots of other things we associate with being mature and healthy) and the limbic system (which is associated with raw emotions that can be overwhelming and "messy" if not regulated) are strengthened. So in a very real way psychotherapy helps your brain use the "smart part" (frontal lobe) to regulate your more primitive emotional center. This give you more control over intense emotions that otherwise may derail you from staying balanced.
The bottom line here is that our brains do change. Even in adulthood. This is good news for those of us who would qualify as "old dogs"! So if there are things about yourself that you wish were different I would encourage you to consider psychotherapy. As one person put it, "it's never too late to have a happy life".