Middle English shrinken, from Old English scrincan "to draw in the limbs, contract spontaneously, shrivel up; wither (through death, age, disease, etc.), pine away" (class III strong verb; past tense scranc, past participle scruncen), from Proto-Germanic *skrink- (source also of Middle Dutch schrinken, Swedish skrynka "to wrinkle"), probably from PIE root *sker- (2) "to turn, bend."
Originally it had a causal form, shrench (compare drink/drench). The sense of "become reduced in size" is recorded from late 13c. The meaning "draw back, recoil" (early 14c.) often was in reference to the behavior of snails; the meaning "flinch, wince, draw back from fear or shame" is by mid-14c. The transitive sense of "cause to shrink, make to appear smaller" is from late 14c.
Shrink-wrap "clingy thin plastic film" used in food packaging is attested from 1961 (shrinking-wrap is by 1959). Shrinking violet "shy person" is attested by 1882.