Hi, English isn't my mother tongue so I was asking myself that question since I first encounted a w/... Back then I was like: "What tf does 'w slash' stand for?" And when I found out I was like "How, why, and is it any intuitive?" But I never dared to ask that until now
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)