Exactly how they passed the 13th, 14th, and 15th amendments. In my opinion, and a known principle in law for centuries, is coercion or duress renders any legal agreement invalid.
According to Law Professor Glenn Reynolds of "Instapundit", this forced ratification of those amendments is much discussed in academic legal circles.
It was then that the radical Republicans in Congress passed the 14th amendment to usurp all kinds of power for the federal government at the states' expense AND to say that Democratically elected politicians in the Southern states could not hold office. The Southern states rejected this and it did not pass.
Then the Radical Republicans went full totalitarian starting the Occupation of the Southern states by military force and disqualifying those elected Southern politicians and disenfranchising practically all the voters in the Southern states based on the argument that the Southern states were "out" of the union so they could do this. Of course, their entire rationale for starting the war is that the Southern states could never leave and were therefore still "in" the union......but if they were "in"....show me the provision of the constitution that allows the federal government to disqualify office holders and disenfranchise voters? There isn't one of course. One of the conditions imposed on the Occupied Southern states was that they had to ratify the 14th and 15th amendments to get back "in" to a union they never supposedly left. All of this was blatantly unconstitutional. The 14th and 15th amendments never legally passed.
and lest anyone think this analysis mine, just go back and look at the legal scholarship on this. There is a long history of recognizing this was all unconstitutional and that these two ammendments are therefore illegitimate.
PS.....though they claimed the Southern states were "out" when they rejected the 14th amendment, they recognized them as still being "in" and those elected politicians in those states as being perfectly legitimate office holders when it came to recognizing their passage of the 13th amendment. They only magically became illegitimate the second they did something the Radical Republicans disagreed with.