Why does "to dip" mean "to leave"?

In AmE "dip" has assumed the present day meaning of 'leaving'/ 'getting out' or, simply, of being vanished in colloquial usage. We know not how it has so evolved or at all it meant as such. What We suggest is a pure guess work.

In our part of the world (India) we take

• holy dip in water
• dip cattle in treated water
• use onion dip
• even dip the flag in honour.

It is this meaning of 'dip' that does the trick. At the time of dipping, what is dipped is removed from the sight. In our day to day parlance, (Street talk) we streach this meaning of 'dip' to its illogical extreme to mean 'playing truant' or staying away from somewhere. English knew no boundaries and perhaps, it will ever know . Probably, it this use of ours that has sneaked into your usages. Who knows!

While also a bit of a stretch, This might answer your question. Dip is a thinner form of tobacco and the phase "Let's dip" could come from people being made to go outside to smoke/chew etc. and not contaminate homes with the overpowering smell.

"Let's go outside and smoke some dip," shortened to "Let's go outside and dip," to, "Let's dip." meaning, to go outside, and finally, to leave.

I'd say this is just an evolution of the word "duck". As in "Lets duck" which means lets get out of here. Ducking and Dipping seems pretty similar and someone creatively decided to use dip instead of duck and it stuck.

I fear we are looking for an etymological reason where we should instead be looking for a behavioral explanation.

In this case, I believe the dip is actually referring to a bow or curtsey, such as would have traditionally rendered before leaving a formal gathering. We also still have common usage of bowing out of things.

Here is the relevant entry for dip as a verb in J. E. Lighter, The Random House Historical Dictionary of American Slang, volume 2 (1994):

dip v. ... 2. to hurry away; DUCK. {1984 quot. perh. reflects an independent use.} 1903 Hobart Out for the Coin 14: He ... grabbed his lid ... and dipped for the woods. 1984 Toop Rap Attack 158 : Dip, or buff: terms for leaving.

The earlier example, from Hugh McHugh [George Hobart], Out for the Coin (1903) reads in context this way:

"Finally he [Uncle Peter] got so rich that he used to trip and fall over the day's winnings when he tried to lock up shop in the evening. He then decided to build a fort around his rake-off, so he grabbed his lid, shook a day-day to the Street [that is, Wall Street], and dipped for the woods."

As you can see, this book is very free with McHugh's representation of Wall Street lingo of the turn of the century ("shook a day-day" seems to mean "bade adieu").

The 1984 quotation is from David Toop, The Rap Attack: African Jive to New York Hip Hop (1984):

Breakout, dip, or buff: all are terms for leaving. Example: 'Fellas I'm gonna breakout because I have to meet my woman.'

Tom Dalzell, Flappers 2 Rappers: American Youth Slang (1996) identifies dip as one of many terms in 1980s and early 1990s "hip hop & rap" slang used to signify that one is leaving:

To leave is to Audi 5000, bill, blow, book, boogie, bounce, break, break north, buff, bux one, clock out, dip, flex, ghost, haul ass, jet, motor, outie, parlay, step off, swayze, or tear up.

Dalzell notes that in the hiphop slang of the same era, dip (along with dap and dope) could also mean "Good with a fashionable twist," The term dip then crossed over into white/interracial U.S. youth slang in the 1990s, as Dalzell memorializes in this exceedingly brief entry:

dip To leave

And finally, Tony Thorne, Dictionary of Contemporary Slang, fourth edition (2014) has this for dip:

dip (out) vb American to depart, leave. A vogue term from black street slang of the 1990s. The variant form 'do the dip' has also been recorded. A variety of euphemisms (like its contemporaries bail, book, jam and jet) for 'run away' are essential to the argot of gang members and their playground imitators.

None of these sources suggests how the word dip (in the sense of leave) arose in the first place. I share Lighter's skepticism that the 1980s hiphop use drew directly from Wall Street jargon of 80 years before, particularly since none of the slang dictionaries covering the period in-between record any contemporaneous use of dip in the relevant sense.

Having said that, I can imagine that either the 1903 Wall Street dip, the 1984 hiphop dip, or both, independently, might have arisen simply from a truncation of the verb depart. Unfortunately, this is pure speculation in my part. I haven't found any reference work that attempts to link the slang word dip to the well-respected standard English word depart.

Dan Bron is correct. My mother from the 60s even knew how to dip out! My generation... we used dip or dipping with a nod of the head to signal leaving. I graduated in 1994. Midwest. Maybe my friends and I were weird... but it was our signal system.