u/J-Force and u/Coeurdelionne might have more to say about this - for some previous recent answers to similar questions, see J-Force's, How did Richard I come to be so fondly lionized in British cultural memory given how marginal of a King he seems to have actually been?, and Richard I of England paid a huge ransom to the Holy Roman Emperor, how did that impact the HRE? by Coeurdelionne.

For your question more specifically, Richard was captured because he had offended relatives of the Holy Roman Emperor during the Third Crusade, and supporters of the Emperor caught him while he was trying to get back home without being noticed.

The very brief background is that during the Third Crusade, the king of Jerusalem, Conrad of Montferrat, was killed by two Assassins on April 28, 1192. The Assassins were captured and executed, but first they claimed that Richard had hired them. It’s not really clear what Richard would gain from having the king murdered, so it’s more likely that the Assassins were trying to destabilize everything on both sides (they were enemies of Saladin as well as the crusaders). But some people believed Richard really was at fault.

Meanwhile Richard had also offended the other leaders of the crusade. Philip II of France (coincidentally also Conrad's cousin) had already returned home in 1190, since he couldn’t get along with Richard. Richard also offended Duke Leopold V of Austria (another one of Conrad’s cousins) by refusing to allow Leopold to fly his flag alongside Richard’s. Conrad, Leopold, and Philip were also all cousins of the Holy Roman Emperor, Henry VI.

Richard and Saladin eventually settled a treaty to end the crusade. Richard's army broke up at the end of the crusade and everyone either stayed in the east or sailed back home however they could. So he wasn’t sailing back home with a big fleet, like he had been on the voyage east. He also had to find a route home that wouldn’t pass through territory of his new enemies. Sicily was part of the Empire, so he risked getting caught if he sailed that way; if he made it past Sicily and sailed up the western coast of Italy, he would also eventually run into Montferrat territory, which certainly wouldn’t be safe, or southern France, also unsafe. So the best option seemed to be to sail up the Adriatic, and pass through Hungary and then to Bavaria. The Duchy of Bavaria was part of the Empire too, but Duke Henry was married to Richard’s sister Matilda, and surely Henry would provide safe conduct back home.

Unfortunately for Richard, his ship got caught in a storm and he landed in northern Italy, within the territory of the Empire. He still tried to reach Bavaria, but he was eventually caught in a little town near Vienna, in Leopold's territory. There are various stories about how he was caught, but he may have been dressed as a pilgrim, or perhaps as a cook - supposedly, when Leopold’s men found him he was cooking a chicken. He was discovered either because he (or his men) had been spending a suspiciously large amount of money, or because he was wearing a ring that was obviously too valuable for the type of person he was pretending to be. The French and German sources that are hostile to Richard tried to portray him as foolishly as possible, and English sources either portray him more heroically or avoid giving specifics about his capture, so it’s hard to know exactly what happened.

In any case, Leopold sold him to Emperor Henry, who then put him on trial for the assassination of Conrad. Richard defended himself well, and supposedly he even managed to produce a letter from the “Old Man of the Mountain”, the leader of the Assassins, that said Richard had nothing to do with the murder. Presumably he forged the letter, but it was convincing! Henry dropped the charge of murder, but Richard was still held prisoner in the Empire until 1194.

At one point he may have been humiliated by being forced to wear chains, but for the most part he was treated well. He was held at Trifels Castle at first, in the Rhineland, and then moved to nearby Haguenau. He was able to have visitors from England and France, including English negotiators, but also ambassadors from Philip, who mocked Richard's inability to defend his lands against Philip’s invasions. Apparently on several occasions he thought he might be killed, or otherwise die in prison, since no one was sure if anyone would ever be able to afford the ransom. So the conditions of his imprisonment probably weren’t excessively harsh, but they were still unpleasant. While he was there he wrote a well-known troubadour song, “No man who is imprisoned” (“Ja nus hons pris”).

His mother Eleanor of Aquitaine arrived in 1194 and helped negotiate the ransom. Richard was finally released in February 1194, for an enormous ransom of 150,000 marks.

Sources:

John Gillingham, Richard I (Yale University Press, 1999).

John Gillingham, “Coeur de Lion in Captivity”, in Quaestiones Medii Aevi Novae 18 (2013).