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:
- Coruscant shall be relinquished to the Republic (no other named worlds or systems are mentioned)
- Imperial forces shall stay within the borders defined by the treaty
- There was also a clause on trade that gave the Republic the upper hand or dominating influence on Imperial trade policy, probably to further the Republic's goal to restore galactic trade to pre-Empire levels while avoiding all the mess that plagued the last years of the Galactic Republic
Clauses that demonstrate their true priority - reducing the Galactic Empire's military threat potential with minimal non-military side effects:
- All Imperial military academies shall cease operations
- The Imperial Stormtrooper Corps shall disband
- The Galactic Empire is banned from operating new and current capital ships to immediate effect
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:
- Parts of the galaxy are still under Imperial rule by treaty
- The First Order's actions are a violation of the treaty
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:
- The Galactic Empire appeared to have fractured into factions after the treaty. By TFA, is the First Order one of those factions or have they "won"? After all, it had never truly been mentioned or implied that the Galactic Empire has officially ceased to be.
- Is the First Order even one of the warring factions to begin with? This is a new scenario I just thought of after reading the article on the treaty - can the First Order possibly be a splinter faction formed by those who refused to recognize the treaty three decades ago, founded by those who jumped into the Unknown Regions back then, and whose territory may thus be established in the Unknown Regions as opposed to the known galaxy? (Therefore mirroring Legends in the form of Thrawn's Empire)
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.