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I'm looking for educational resources that I can share with others in an effort to engage more citizens in how our election system actually works.
Why does gerrymandering exist, and how does it actually work?
I've done a lot of online research and have spoken to many individuals who are knowledgable on the subject, but can not find actual trusted educational documentation that I can share with the general public.
First, to learn more about origin: Start with Wikipedia; and then read the Atlantic's reasonably in-depth coverage here. Then go on to FiveThirtyEight's articles on the topic. You are welcome to read Washington Post's article linked in another answer, but whether you consider WaPo "trusted" largely depends on your political leaning - those who align with WaPo politics trust it, those who don't, do not.
As a side note: the main reason gerrymandering exists is because the congressional districts are allocated by people. Politicians. Who have motivations.
Now, people often attribute different motivations to gerrymandering, but the main motivation of politicians is always one: gain and retain power. That comes in two flavors: winning an election in the first place; and being a part of a party that has a majority once you won an election. Note the first part - if you don't get to win, it doesn't matter who's in the majority, as you lost the seat.
Two important factors should be noted when researching gerrymandering as a current process:
First, the fact that it doesn't actually have all that much of an effect.
A study "Evaluating partisan gains from Congressional gerrymandering: Using computer simulations to estimate the effect of gerrymandering in the U.S. House" by Chen and Cottrell, published in An International Journal on Voting and Electoral Systems and Strategy, puts paid to the frequently asserted but rarely proved idea that somehow, the main goal and main effect of gerrymandering is to increase the party's majority, by showing that the net affect of gerrymandering in US is negligible:
The analysis reveals that while Republican and Democratic gerrymandering affects the partisan outcomes of Congressional elections in some states, the net effect across the states is modest, creating no more than one new Republican seat in Congress. Therefore, the partisan composition of Congress can mostly be explained by non-partisan districting, suggesting that much of the electoral bias in Congressional elections is caused by factors other than partisan intent in the districting process.
FiveThirtyEight was analyzing this from a different angle but also does not find that gerrymandering as as much effect as usually claimed:
A variety of academic analyses of redistricting have found that this geographic self-sorting accounts for much — probably most — of the “skew” of Congressional districts against Democrats. Gerrymandering and other partisan efforts at redistricting do play a role, but it is mostly around the margin. A study by John Sides and Eric McGhee found that redistricting after the 2010 Census, which was controlled by Republicans in many key states, produced a net swing of only about seven House seats toward Republicans.
Second factor is gerrymandering can be self-defeating (and this self-moderaing). That is due to the fact that the two sub-motivations mentioned above - desire to win individual seats, vs desire to win as many seats as possible - often conflict with each other (538 covered it in detail, i'll try to find a link).
If you redistrict to maximize voting efficiency for your party (thereby by optimizing for increasing the total # of seats your party gains, in theory), that automatically means there's less vote margin for each individual congressperson, meaning they are more likely to lose - and that doesn't even count the fact that in a wave election, you're liable to lose most/all of the thin-margin "efficient" seats. As a matter of fact, that factor did backfire on gerrimanderer's in certain elections.
I can only answer part of the question, but this picture is the best, easiest to understand explanation of how gerrymandering works that I've seen. I think it's a little sad that the original (and this adapted version) use blue and red, instead of two neutral colors, but it generally reflects the reality.
