The development story of each of those languages is different, so I'm going to confine this post to broad strokes and specifically Yiddish.

First, the broad strokes. There are two terms usually used for "mixtures" of language: pidgin and a creole. A pidgin is developed when two different language speakers need a means to communicate. What that generally entails is a borrowing of words from each other and cobbling together a grammar that facilitates communication. Anything that's difficult in either language is stripped away (e.g. clausal phrases, verb inflection, etc.). Additionally, a pidgin is not stable and continues to develop over time, usually in the same haphazard way it developed in the first place.

A creole is, to put it simply, a pidgin that children are raised to speak natively. That means the language is going to be more stable and feature all the simplified structures of the pidgin. But, because there are now native speakers of our impromptu language, that means they're going to start playing with it and necessarily adding in deeper structures that we, the parents trying to communicate, could not use. These structures will develop naturally, just like the recent developments in English with "because" being used as a conjunction (e.g. "because science," "because reasons"). The children will use their native language in hip new ways that make the parents cry ai-yi-yi, just like they do today.

So, many of these languages developed out of necessity to communicate with the locals. Jews are stereotyped as business owners, so they would have enough prestige in many communities to donate some words to a pidgin instead of just adopting the local language. (Look at how few words an oppressor's language adopts compared to how many words the oppressed's language adopts, e.g. English and French.) That's only part of the (generalized, broad-strokes) story.

Jews also take their language and religion with them wherever they go. They necessarily need Hebrew or Aramaic, or some form of it, to read their scriptures and worship Yahweh. This is a big factor in getting Hebrew mixed up with other languages - a cultural necessity. (Remember, this is broad strokes.)

Yiddish, in particular, is theorized to have been a High German language that borrowed copiously from Hebrew and Aramaic (this is the mainstream view) and later borrowed Slavic too, but for whatever it's worth, Max Weinreich (History of Yiddish, 1980) argues that it developed from French Jews picking up German while relegating Hebrew/ Aramaic to religious contexts, and only later did the Slavic get slipped in and the Hebrew/ Aramaic slip out of religious contexts and into everyday life. I wasn't convinced by his arguments, but I also don't have the credentials to dispute them intelligently. We have evidence of Yiddish tracing back to the 12th century, so we never saw it as a creole (rarely do those get written down), but we know that it ("the mother tongue") was referred to differently than Hebrew ("the holy tongue") and that German was also referred to differently (taytsh Yiddish vs. tiutsch Middle High German).

Lastly, I want to mention that whether a language is defined as a pidgin, creole, etc. depends on who you talk to. Some will liberally apply the term creole so that it include languages such as English after the Norman invasion, others won't. A pidgin is also hard to define exactly what it is. How many words do we have to borrow from each other before it's a pidgin of these two languages instead of German with a couple bits of Hebrew? If you think biologists have it hard defining a species, get a group of linguists to argue about what borders define a language! Even my professors in my graduate work had different views on the limits of pidgin and creole.

So, a few ways. First, jewish languages tend to borrow heavily from Hebrew and other jewish languages. This happens because jewish languages are well know in jewish communities, and because for some terminology there's no native word in the local language. This happens in Jewish english, too.

Also, there's gradual divergence that took place. Regular sound change is pretty common. Segregation and/or geographic separation from non-jewish speakers could facilitate this, but there it does observably happen in American English spoken by Jews, too.

A third bit is geographic separation causing new loans. One way Yiddish is different from standard German is its inventory of Slavic loans. The same happened with ladino in the Ottoman Empire.

So, these languages tend to be differentiated with loans from other jewish languages, and by divergence and loans that grows over time.

By coincidence, a past episode of the AskHistorians Podcast covers this very subject. I've PM'd /u/gingerkid1234 to see if he would like to chime in.