What does the punctuation "//" mean?

// comes from programming and is generally used to denote a comment or explanation that should be ignored by the compiler or computer. Its purpose is to leave notes and instructions for future programmers or anyone else that needs to understand what is happening in code. The literary equivalent is a footnote.

In the examples you provide, the words following the // are comments of emotions or actions to let the reader know what the author was feeling or doing.

I think I owe myself a THWACK. //ashamed

This means that whoever said this was ashamed of their actions. Similar constructions include using :emotion: or an emoticon:

Yeah, that is great idea. :eyeroll:

This is great news! :)

Other programming style behaviors include ^H and nested or malformed ( ) [ ] { } pairs:

I will stalk^H^H^H^H^Hfollow you everywhere!

I like pizza (with olives (except green olives (but you can put them on your half of the pizza))).

I know this is not relevant to your case, but a single or double slash is also used when citing poetry, to indicate where the metre or line ends in a continuous quotation: at regina gravi iamdudum saucia cura // vulnus alit venis et caeco carpitur igni. It looks like this in proper print:

At regina gravi iamdudum saucia cura

Vulnus alit venis et caeco carpitur igni

It is very informal, looks like a variant of an emote which uses a singular slash, the double is probably influenced by programming comments, but hard to say. Particularly as a double slash in written work usually means "new line here".

In the UK, a double slash // is often used to denote where a new paragraph or line should have started.

This is often common in schoolwork, where students may realise they should have started a new paragraph after-the-fact, and must add it in later without re-writing a significant portion of their work.