At the end of Ladder games, your skill level is reevaluated based on the outcome and the level of your opponent(s).
Your level is represented by a number and rank.
Your skill level's number is shown below your icon. Your rank can be either bronze, silver, or gold within that level. This is shown as the background behind your icon.
You start a level at bronze rank and advance through silver and then gold. When you advance past gold rank, your skill level number is increased. Unfortunately, if you don't do well, you can un-advance down the ranks and down the skill levels.
At the end of a game, if your level or rank has changed, it will be shown as a part of your scoring summary.
In the above image, Marie has advanced from level 2 silver rank to level 2 gold rank.
Your skill level is used to find competitive opponents during Ladder and Casual matchmaking.
As your skill level increases, the number of wins it takes to advance increases. So, if you win and don't see a change, don't worry, the system is keeping track of it!