All in all, the / is just one style of abbreviation used in English. It's not only used for "with", but also a few other words (w/o = without, N/A = not applicable).
In German we abbreviate using a dot (e.g. "m." = "mit" = "with). That's not more or less intuitive, it's just what you are used to.
What's kinda special with English is that there are multiple abbreviation styles. Off the top of my head I can think of six styles:
- Abbreviate random parts of words using a slash: "N/A", "w/", "w/o"
- Abbreviate keeping only the first letter of a word using a dot: "e.g."
- Abbreviate keeping the first and some random later consonants (and sometimes consonants that aren't in the word at all) without using punctuation: Dr, Mr, Ms, Mrs
- Abbreviate using acronyms and no punctuation: BBC
- Abbreviate using acronyms and dots: B.C.
- Abbreviate by substituting parts of the word with a single letter: Xmas (Christmas), Xing (Crossing)
That's why piracy is back on the menu.