It was introduced by the philosopher and logician Willard Van Orman Quine in his book Mathematical Logic, originally published in 1940.
Quine had hoped that, by avoiding variables and schemata, he would minimize confusion for the readers, as well as staying closer to the language that mathematicians actually use.
Suppose, for example, that one wants to define the well-formed formulas (wffs) of a new formal language, L, with only a single logical operation, negation, via the following recursive definition: Interpreted literally, rule 2 does not express what is apparently intended.
Quasi-quotation is introduced as shorthand to capture the fact that what the formula expresses isn't precisely quotation, but instead something about the concatenation of symbols.
The expanded version of this statement reads as follows: This is a category mistake, because a number is not the sort of thing that can be concatenated (though a numeral is).