The number of stars on your minimon is it's grade. You can increase the grade in several different ways. On of the most common ways of increasing the grade of a monster is through random evolution.
Random evolution
Random evolution combines 2 random minimon at max level and upgrade, to make a random minimon that is 1 star higher. This is a good evolution type to use for fodder monsters that aren't very useful. Just make sure to read their abilities to see if they are worth keeping before using them for random evolution.
Fixed Evolution
Another way to increase the grade of your minimon is through fixed evolution. This involves evolving a max level and upgrade minimon with an evolution stone of the same type and grade. You get these evolution stones from the daily dungeon.
Here's an example of a fixed evolution.
Evolving evolution stones!?
You might be asking yourself, "How do I get my 1 star evolution stone to 2 stars?". Good question, you can increase the grade of evolution stones the same way you would for monsters. Except you don't have to level evolution stones. You can increase the grade of these stones in the "Evolution Stone" section in the bottom right corner of the character panel.
Want to know how to evolve your hero to a mutant? Check out this blog post specifically about mutant evolutions.
