Enlighter has three different methods to mark code blocks to be highlighted. These methods are related to the editor mode your are using!

Visual Editor Mode

Enlighter provides a full integration into the visual editor (Gutenberg+Classic)! You can easily add code snippets by using the Code Insert Dialog located in the upper right corner of the editor toolbar.
By focusing a code block in the visual editor, a edit icon will appear - this allows you to change the preferences of this snippet.

This mode allows you to use Enlighter without any worries. But take care: do not copy+paste snippets directly in the editor!

It is not recommended to use shortcodes and/or markdown syntax in this mode, because the editor will interpret this sections as regular content and is adding p/br tags into your code.

Text/HTML Mode

In this mode, you are allowed to post plain text/html. There are no additional tags which are added to the editor. The recommended editing methods for this mode are:

Shortcode like [php]echo "Hello World";[/php]
Markdown like `echo "Hello World";`