Start by changing the font of the text. You want 1-2 fonts across your entire game UI.
Scale the text so that it fills an appropiate amount of the box.
Make a bar effect, so that when you gain exp, there is a blue bar that moves from left to right (or down to up). You do this by increasing the X size of the blue bar as exp increases. And then tweening it so that it looks good when the player gains a lot of exp.