Basically, this is said as an entry-level phrase to someone you have a favorable
impression with, or if you are a very 'traditional' person. Another situation you would use this, is if you were being convinced to accept someone's
hospitality. I am unfamiliar with the direct transliteration, but if you wash the grammar properly it comes out to be 'Please treat me kindly' or 'Please take care of me' or 'I am in your care, kindly' something to that effect. As you can see, this phrase is a trade of trust, where you 'ask' for trust, and 'give' trust by 'placing' yourself in someone's care.
Source? My frequent visits to Japan, and
familiarity with its culture.
'Yoroshiku onegai shimasu, senpai.'
- Here, an underclassman at school or work who might have
previously met this person they refer to as 'senpai' (or senior,
upperclassman) Implores them to continue their goodwill now, and in the future. The meeting may be in front of the senior's friends, so the person wants to be
respectful so their senior doesn't lose face.
