Destacar (Spanish for “to stand out”) comes from the French destachier (“to detach”) which, in turn, comes from the Latin de- (of, from) plus the old French stakon, meaning a “stake” (literally, as in a pole!).
Thus, “standing out” (destacar) is literally just detaching yourself from the rest around you — who are, presumably, much lower quality than you are!
We can see the root clearly in the d-(s)-t-c (for destacar) to d-t-ch (detach) mapping.
Don’t forget that the de- prefix in French and sometimes Spanish is just another form of the de- prefix. Thus, explaining the extra -s-. And — clearly! — attach comes as well from the same root, just without the de/des negation!
But the best modern English word from the same root is… staccato. Yup: playing the piano in staccato fashion is just, when you play each note really separated from the others!