The apparent impersonal use of the third person (interest, interfuit etc.) is a contraction of inter rem est, for something like 'it's within the affair [of]' : the contracted form is thus misleadingly seen to be a part of interesse.
Although other constructions are used, [interest] is primarily followed by a genitive referring to the actual res when necessary, e.g. interest verborum for 'the phrase is interesting'.
The derivation from inter rem est is given in Dr. Smith's Latin-English Dictionary, at article IV under the heading 'Intersum'. It is not always given in other reference works, but it seems (to me) to be implicit in them as they very often follow Smith quite closely.This is especially so in the following (although it actually preceded Smith).
The oldest article I can find on the usages of impersonal [interest] is in the'Copious and critical Latin-English Lexicon' of J.E.Riddle (1841). It also is, I think, the best: it certainly seems that his article on intersum, with its analysis and examples, has formed the basis for everything subsequent that I have looked at, including Smith's (which, like Riddle's, is is based on Freund's earlier work and several others), 'Lewis & Short' and the familiar primers and instructional books (Bradley's Arnold, Hardie, Kennedy, North & Hillard etc.). It does not actually explain the formation (as does Smith — see above) as a contraction from inter rem est, but appears to assume that it is correct.
Under the heading INTER-SUM, Riddle considers in art. IV the impersonal use of interest in a lengthy article with four sub-headings, distinguishing and explaining various uses, with examples from the classical authors, extracted below.
(A) With genit. of the person to whom any thing is of importance; but if that person be represented by a pers. pron., the abl. fem. mea, tua, nostra, vestra is sometimes used, or cuja (for cujus) — quoting Cic. Q. Fr. 2, 4, followed by a series of illustrations. [followed by] Instead of a person to whom the thing itself is of importance to any one may be added by the preposition ad : interest ad laudem civitatis, Cic. N. D. 1, 4 —ad properationem meam quiddam interest non te exspectare.
(B) The degree how much any thing is of importance is also expressed by neuter adjectives : multum interest, te venire, Cic. Fam. 12, 9; . . . . —ad disciplinam militiae plurimum intererat, militem insuescere, Liv. :—quid illius interest, ubi sis? What does it matter to him? — or by an adv., as, maxime, quantopere : quantopere intersit, opprimi Dolabellam. — By a genitive of value : illud magna mei interest, ut te videam, it is of importance or consequence to me, Cic. Att. 11, 22 [. . . . further examples].
(C) The matter which is of importance may be expressed by an inf. or acc. and inf., or by a final or interrogative clause, introduced by ut or ne, or by an interrogative particle [followed by examples including:] quantum salutis communis intersit, duos consules esse, Cic. Mur. 2 — vestra interest, ne imperatorem pessimi faciant, Tac. —Seldom by a subst. : in Epirum statui me conferre, non quo mea interesset loci natura, Cic. Att. 3, 19, 1. — more frequently by the neuter pronoun :vestra hoc maxime interest, it concerns you most, itis your affair more than any one's else, Cic. Sull. 28.
(D) Non tam interest, quo animo, Caec. in Cic. Fam.6, 7.
This should at least satisfy the concerns expressed in comments made before this addition was in place.