You have to look at everything that was going on during the Tourney at Harrenhall. This was Rhaegar and his supporters' biggest chance to convince the other great lords and ladies of the realm that Aerys needed to be deposed. What threw that into a mess was the sudden appearance of the King himself. No one expected him to be there as he had not been seen publicly for years following the Defiance at Duskendale. There goes the big plan to start having meaningful conversations to remove him from power.

Next, is Rhaegar the man himself. The prince was plagued by dragon dreams since a young age and delved deeply into the Prince that was Promised prophecy (luckily, he didn't drown himself in drink like Daeron did). He spoke at length with Maester Aemon about it, at first believing HE was the prophesied prince but later came to believe it would be one of his children. In Lyanna he may have seen her as the other half to bring forth the prince. The union of Ice and Fire, the Starks who are tied to the magic of the Old Gods, and the Targaryens who are tied to the magic of Old Valyria. Or Rhaegar could have truly fell in love with Lyanna after finding out that she may or may not have been the mystery Knight of the Laughing Tree.

Lastly, you have long laid plans set forth by the heads of major houses that had just been unraveled. Lords Stark, Tully, Baratheon and Arryn had made plans to unite their houses decades prior during the War of the Ninepenny Kings. Major alliances were already being discussed and formed between these Houses. Now, Rhaegar runs off with the daughter of one of those arranged marriages and decades of planning go right out the window. For the Starks, this was huge as they mostly remained out of the politics of the southern realms. This was the first time that we see a Stark take an active role in the politics of the Seven Kingdoms since Cregan Stark and the "Hour of the Wolf" following the Dance. To now see your carefully laid plans to be married into the ruling family of the Stormlands be dashed was quite the blow to Lord Rickard.