How do I run an Applet within Intellij IDEA?

The easiest way to run an Applet in IntelliJ is by creating an Applet Run/Debug Configuration. Just click on the little dropdown that shows the name of your main class to execute, click on "Edit Configurations...", then click the green + button, then choose "Applet" and specify your LineSimApplet as the target Applet class.

The reason why your main method approach is not working is because Applets are not meant to be run as standalone applications, but if you do want to run it that way, the easiest way would be to add it to a JFrame.