Most famous/touristy places in India have a story behind their existence. Murudeshwara is no exception.
Long story short: Ravana wants to attain immortality by getting the Atma-Linga from Shiva, which he gets eventually after severe penance. Shiva hands over the same with the only condition being that the Atma-Linga should not be placed at any place before Lanka, and if placed, it would be impossible to move it again. Fearing the disastrous consequences of Ravana possessing the Atma-Linga, Lord Vishnu with the help of Ganesha decide to do something about it. So, when Ravana is near Gokarna, Lord Vishnu blocks the Sun to make it appear like dusk, right then, Ganesha in the disguise of a Brahmin boy offers Ravana that he can carry the Atma-Linga till Ravana finishes his evening prayers, and promises Ravana that he would call his name thrice before he places it down. Ganesha, does call Ravana’s name in quick succession and places the Atma-Linga on the ground where it takes root. Vishnu, unmasks the Sun and its daylight; Ravana realises he is been tricked and gets angry, he tries to uproot the Atma-Linga and in the process breaks the same into few pieces. Finally he throws the cloth covering the Atma-Linga in a place which is called as Murudeshwar.
Wikipedia
This 20 storied Raja Gopura has a lift which can take you to the top floor for getting a better view of the Shiva statue.
Full view of the 123 feet Shiva statue with smaller statues of Ravana and Ganesha (in the guise of a boy).
Ravana handing over the Atma-Linga to Ganesha (disguised as a boy).