One source of its appeal is that it reinstates much of what was ruled out as cognitively meaningless by logical positivism but without rehabilitating the commitments of old-time metaphysics. Permit me to explain. (My ruminations are in part inspired by Ernest Gellner, to give credit where credit is due.)
Crudely put, as befits a crude philosophy, logical positivism is just Hume warmed over. The LPs take his famous two-pronged fork and sharpen the tines. Hume spoke of relations of ideas and matters of fact, and consigned to the flames anything thing that was not one or the other. In the Treatise of Human Nature, he spoke of "school metaphysics and divinity" as deserving of such rude treatment. Since Hume's day, old-time metaphysics and theology have had a forking hard time of it.
The LPs spoke of two disjoint classes of statements and maintained that every cognitively meaningful statement must be a member of the one or the other. The one class contains the truths of logic and mathematics and such analytic statements as 'Every cygnet is a swan' all interpreted as true by convention. The other class consists of statements empirically verifiable in principle. Any statement not in one of these two disjoint classes is adjudged by the LPs to be cognitive meaningless. Thus the aesthetic statement, 'The adagio movement of Beethoven's Ninth exceeds in beauty anything Bruckner wrote' is by their lights not false, but cognitively meaningless, though they generously grant it some purely subjective emotive meaning. And the same goes for the characteristic statements one finds in theology, metaphysics, and ethics. Such statements are not false, but meaningless, i.e., neither true nor false.
Imagine a debate between a Muslim and a Christian. Muslim: "God is one! There is no god but God (Allah)!" Christian: "God is triune (three-in-one)." For an LP, the debate is meaningless since theological assertion and counter-assertion are meaningless. The assertions are neither analytic nor empirically verifiable. Or consider a debate between two Christians. They are both Trinitarians: there is one God in three divine Persons. But the man from Rome maintains that the Holy Ghost proceeds from the Father and the Son (filioque) while the man from Constantinople maintains that the Holy Ghost proceeds directly from the Father. For an LP, this debate about the procession of Persons is cognitively meaningless. I chose these examples to show how attractive LP is. For many of you will be inclined to think of these debates as in some sense meaningless. "How could one know one way or the other?" Many of you will be inclined to want to tie meaningfulness to empirical verifiability. Nevertheless, Logical Positivism is untenable. But that is not my present point.
My present point concerns the appeal of OLP. The OL boys weren't out to resurrect metaphysics. They took on board the anti-metaphysical animus of the LPs. But their approach allowed the salvaging of ways of talking that the LPs had no interest in preserving. Religious language is a key example. So what I am contending is that one source of the appeal of OL philosophy was that it allowed religious talk and thus religion itself to be saved from the forking accusation of meaninglessness. But it did this without crediting old-time metaphysics. You can see why that would appeal to a lot of people. To explain this properly would take a lot of scribbling.
But the central idea is that religion is a form of life and a language game, a self-contained language game that needs no justification ab extra. Hence it needs no justification from metaphysics or philosophy generally. It is in order as it is -- to use a characteristically Wittgensteinian turn of phrase. By the same token, religion cannot be attacked from the side of philosophy. It is an island of meaning unto itself, and is insofar forth insulated from criticism. (L. insula, ae = island.) Nor can it come into conflict with science or be debunked by science. Within the religious language game there are valid and invalid moves, things it is correct and incorrect to say; but the langauge game itself is neither correct nor incorrect. It just is. Religion is a groundless system of belief, a system of belief that neither needs nor is capable of justification. Since I reject both LP and OLP, I am not endorsing this view of religion. I am merely explaining one of the reasons why people are attracted to OLP: it allows them to practice a religion while ignoring both the threat from traditional philosophy (which demands the justification of key religious tenets) and the the threat of positivism which makes positive science the ultimate arbiter of reality.
This post truncates a larger discussion to be found in What is Right and What is Wrong in Wittgenstein's Philosophy of Religion.