I think your question itself rests on a whole bunch of assumptions that aren't really true:
Japan is unusually weird and perverse when it comes to sex
US/Europe are the opposite.
Being sexually demure and modest is the "normal" condition. Anything that deviates from this is "abnormal."
Culture is frozen in time, unchanging, and uniform across an entire society. Culture, in effect, has no history.
Having "liberal attitudes" about sexuality is specific enough. No need to explain further.
The very way the question is framed smacks of Orientalism. I personally don't think Japanese culture is THAT much more sexually "liberal" than other countries. It's just that foreigners sometimes fixate on odd extreme examples of Japanese sexuality, and then expand it to stand for mainstream "Japan." Actually, probably Japan and Western Europe have similar attitudes toward sex. It is the United States, with its schizophrenic approach to regulating and commercializing sex, that seems more the outlier.
I think "liberal" might be a loaded word. I guess what you mean is permissiveness to sexuality in the public sphere?
The permissiveness to sexuality in Japan developed from Buddhism. It began as early as in the 8th century (the Heian period). It sees sexual pleasure not as taboo as it does in, for example, Abrahamic religion which lies the foundation on Europe. Sexual pleasure instead is seen as a gift that should be appreciated and enjoyed. There is one interesting tale on a monk and a nun in Kokon Choumonju, I quote (some parts are omitted) from Janet Goodwin's The Sex Trade in Heian-Kamakura Japan,
Not long ago, there was a nun who had lived her whole life as a virgin. Once when she was visiting a temple, a certain monk saw her, and he realized that he was helpless before her irresistible beauty. Since there was no way he could overcome his feelings, impelled by a desire that none could imagine he paid a visit to the nun’s house. The monk, being quite feminine in appearance, disguised himself as a nun and knocked at the woman’s door. Claiming to be a widow without means of support, he asked to be taken into her service. Without hesitation, the nun agreed to employ him. He worked diligently as a servant for a year and commended himself to his employer who, still thinking he was a woman, invited him to sleep beneath her robes in winter because the evenings were so cold.
From the beginning of the new year, his mistress the nun devoted herself to special nenbutsu services in her private chapel, secluding herself for seven days. On the eighth day she emerged from the chapel exhausted and fell asleep. Deciding that it was time to bring matters to their conclusion, he forced apart the legs of the sleeping nun and penetrated her. Shocked and bewildered, she jerked away from him and fled into the chapel. Just as he was about to conclude that his efforts had resulted in disaster, he heard the nun cry out, “Where are you?” — apparently not in too bad temper. Overjoyed the monk answered, “I’m right here,” and she emerged from the chapel to find him.
When he asked her, “Why did you push me out the first time and run into the chapel?” she answered, “Because I thought, why should I enjoy such a good thing all by myself? I thought I’d give a portion to the Buddha, so I ran inside to ring the bell for him.”