If you want to learn a language perfectly, the best way to start is to pick it up as a child from your family and the community they're part of. That way, you and your generation end up speaking the same language as your parents and their generation, modulo a few little innovations you threw in just to annoy them. As those little innovations pile up, generation on generation, sooner or later you end up speaking something that the first generation wouldn't have been able to understand. In such a scenario, everyone agrees, the latest generation's language – let's call it B – is descended from the first generation's (A). If some of the children of that first generation moved far away early on and went through the same process of gradual change, their descendants speak another language, C, which speakers of B can't understand, but which is also descended from A. So we say that B and C belong to the same language family, just as their speakers belong at some remove to the same extended family.
If you're reading this, it's probably too late to learn a language that way. (Sorry.) You can still learn another language, say B, but the odds are that, at best, you'll always speak it with a bit of a foreign accent, and keep using expressions that make sense in English but sound weird to native speakers. If you're just an individual migrant learning it to fit in, that won't matter in the long run – your kids will learn the language in the playground and come back speaking it better than you do. But what if it's not just you that's learning it, but also your spouse, and your brothers, and almost everyone you know? What if your whole community is starting to prefer to speak this language with their kids, instead of the one they grew up with? In that case, the kids will still end up speaking it – but instead of speaking it like natives, they'll probably end up speaking it with your foreign accent and all those expressions of yours that native speakers laugh at. In that scenario, does the kids' language (let's call it D) belong to the same language family as B and C, or not? That's the ambiguity that Wexler is playing with.
The obvious answer – and the one most linguists would give – is yes*. For one thing, assuming you did a half-decent job of learning B, it's the same language – speakers of D can understand speakers of B, and vice versa, even if they laugh at each other's crazy accents. The influence of Gaelic may pervade Irish English, but Irish English is still English, not some Celtic language. It's the vocabulary and the morphology that really make English understandable – a weird accent or a funny way of putting things is just not that big an obstacle on its own. Wexler proposes exactly the opposite criterion: "Yiddish – in contrast to its massive German vocabulary – has a native Slavic syntax and sound system – and thus must be classified as a Slavic language" (1993:5). The origins of Yiddish syntax and phonology I can't comment on, but there's a good reason why historical linguists normally prioritise the vocabulary and the morphology over the syntax and phonology, even apart from the one just given. Vocabulary and morphology are eminently reconstructible, using the comparative method. Phonology, on the other hand, can only be reconstructed from vocabulary, and syntax is notoriously hard to reconstruct at all. If language families were to be defined based on phonology and syntax, it would hardly be possible to define them, much less reconstruct them or state regular correspondences between them.
In short, saying that Yiddish (much less Modern Hebrew) belongs to the Slavic language family is just a word game – in the sense that historical linguists normally use the concept of "language family", it doesn't, and wouldn't even if every last Yiddish speaker happened to be of Slavic ancestry and to speak Yiddish with a heavy Slavic accent. But such word games do not vitiate Wexler's work. After a large enough community has shifted to a different language, it is usually possible to find traces of their former language – although identifying them as such, rather than as later borrowings, may be hard. That's what Wexler is trying to do for Yiddish, and that's how he supports his claim that Yiddish speakers' ancestors used to speak a Slavic language.
* However, the question can easily be made more controversial. Suppose you and your community didn't learn it that well to start with, and aren't trying to imitate native speakers anyway? In that case, the kids will end up speaking something that sounds utterly ridiculous to native speakers; the basic words are recognisable, but the way they're put together seems all wrong. Whatever Tok Pisin is, most people would agree that it's not English. A few people would defend the claim that Tok Pisin belongs to the same family as English, on the basis that that's where the vocabulary comes from, but most would say that it doesn't belong to a language family. The language family model presupposes that the language is being passed on reasonably well as a whole, including not just vocabulary but also some amount of grammar; if all that's learned is a bunch of words, the model breaks down. The border must be drawn somewhere between the extremes of Irish English and Tok Pisin, but linguists can and do disagree on where exactly to draw it.