In the case of the second, the context was the horrors of the British militia imposing themselves on the colonists, just a few years prior. The only way the States would ratify the proposed Constitution's clause that the newly-formed government apparatus could raise its own militia, was to add some kind of regulatory mechanism: in that case, the right of private citizens to keep and bear arms, as a check and balance.
In the case of the fourteenth, the context was how to confer newly-freed blacks citizenship, but without simultaneously opening up silly loopholes whereby every foreigner and their uncle could also become citizens. So, the first sentence was added to the amendment, indicating that anyone who was not already subject to a foreign power upon birth-- meaning, coming from parents of a foreign nation-- would be citizens of the United States. According to the clause's own author, compliments of Wikipedia-- bold emphasis is mine:
"While the Citizenship Clause was intended to define as citizens exactly those so defined in the Civil Rights Act, which had been debated and passed in the same session of Congress only several months earlier, the clause's author, Senator Jacob M. Howard of Michigan, phrased it a little differently. In particular, the two exceptions to citizenship by birth for everyone born in the United States mentioned in the Act, namely, that they had to be 'not subject to any foreign power' and not 'Indians not taxed', were combined into a single qualification, that they be 'subject to the jurisdiction' of the United States..."
In addition to the clause's author, I also found quotes from Lyman Trumbull and John Bingham-- both of whom were involved with the amendment from day one, and the latter of whom actually wrote the rest of the text-- that one hundred percent concur with Jacob Howard's explanation above.
But, just as was the case involving the second amendment, activist Supreme Court judges much later on essentially re-wrote the Constitution's text by radically re-defining what the actual words mean, in order to suit their political agendas.
This is getting brought up a lot right now since Trump is considering an executive order, instructing agencies to enforce the original intent of the Constitution's text. If he can get that accomplished, I see that as a win based on what my research has uncovered. Of course, Democrats and their followers will be arrayed against this, because it will take away the "future Democrat anchor baby" pipeline.