There are various newly-published literature that tell us some of what happened after that, Aftermath and Lost Stars among them. The key to your answer lies in the Galactic Concordance. The Galactic Concordance is the peace treaty signed between the Galactic Empire and the New Republic soon after the Battle of Jakku. At the Battle of Jakku, the Galactic Empire attempted to take revenge on the Rebellion - now self-proclaimed as the New Republic - and crush them once and for all, but the odds were turned and the bulk of the Imperial Navy were exterminated in that one battle. Without a means to continue the war strategically (it's like the USA losing all but a couple aircraft carriers in the middle of a war with the USSR), the Empire sued for peace in an effective surrender. Key terms of the treaty relevant to this question are as follows. Clauses that suggest they aren't seeking the Empire's immediate dissolution and are willing to tolerate continued existence:
Clauses that demonstrate their true priority - reducing the Galactic Empire's military threat potential with minimal non-military side effects:
The article on the treaty ends by mentioning that an Imperial fleet jumped off after the treaty was signed for the Unknown Regions (Starkiller Base is later known to be at the edge of the Unknown Regions according to a galactic map in the visual dictionary). The key points to be drawn from the above (some to correct a few points in your question that I think are incorrect) are that:
The above answers one part of "how the galaxy got here" — how did we get from RotJ to TFA, excluding the First Order for now? The next part - addressing the First Order's rise - is as far as I know still speculatory at this point because of the following questions with no documented answer to date:
This is what we know so far about what happened since RotJ. We generally know how the New Republic got to where it is today, but not the same for the First Order. Whether there are other "Imperial remnants", we also do not know, or if the First Order is distinct from any still-existing Galactic Empire.
The New Republic’s rise reduced the once-mighty Empire to a rump state hemmed in by strict disarmament treaties and punishing reparations. But in the galaxy’s Unknown Regions, former Imperial officers, nobles and technologists plotted a return to power, building fleets and armies in secret. This movement became the First Order, ruled by Supreme Leader Snoke, and is now ready to reclaim the Imperial legacy.
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