A self-assertion is a boastful statement. "I am the greatest!" is a self-assertion. Self-assertion is great if you’re a boxer like Muhammad Ali, but usually annoying if you’re not.
An assertion is an opinion like "The Yankees are the best." A self-assertion is a similar claim made about yourself. A low-level politician saying, "I will be President" is making a self-assertion. An athlete saying, "I will win the championship" is making a self-assertion. Self-assertions tend to be a little inconsiderate and arrogant, because self-assertions are usually a form of bragging. Someone who really feels confident probably wouldn't need to make a lot of self-assertions.