This phrase is used both literally, to refer to someone in a kneeling position, and figuratively, to describe someone making an earnest and momentous request or plea. The concept of kneeling to make requests is extremely ancient, especially in Western culture, where people have been kneeling to pray, accept honors, and to ask for favors for thousands of years. In the East, people often go beyond bending, making such requests in a seated position or a deep bow.
You may be wondering, incidentally, why the phrase is “on bended knee,” rather than “on bent knee,” since “bent” is generally viewed as correct English. The use of “bended” is actually archaic; “bent” emerged around the 1600s, but “bended” has been retained specifically in this phrase since then. The term “on bent knee” is also correct, although it sounds slightly less exciting and romantic.
The implication of bending or kneeling is that the person who is bending is submitting. People often adopt bent positions to pray, indicating a submission to God's will. People have also historically bent to mark respect for and submission to people who are of a higher social rank. This has been especially true in circumstances in which people wish to request something, such as a boon of land, titles, or clemency from a lord historically.
When someone says that he or she made a request “on bended knee,” the implication is that he or she asked as humbly as possible, in a way which might be said to be bordering on groveling or pleading. This would also suggest that he or she tried as hard as possible to ensure that the request would be granted, by indicating that the outcome of the request was entirely up to the person with the power to grant it.
There is not necessarily any shame in going to great lengths to make a request, especially when you are dealing with someone who has very traditional or formal values. Taking the steps to lay out a request clearly and politely is always a good idea if you want the request to be granted, and sometimes being especially submissive and deferential to someone can impress them with your earnest desire for them to grant your request. In most cases, you do not need to literally go on bended knee, unless your benefactor has an appreciation for dramatic flair. However, it is possible to go overboard; be careful about seeming obsequious when you request something, and make sure not to cross the line into artifice when making a request.
While people rarely literally kneel to make requests in the West anymore, there is one circumstance in which it is considered appropriate to ask for something on bended knee: a marriage proposal. The retention of the tradition of kneeling in this particular instance is probably related to the long formal traditions associated with engagements and marriage, and many people find it charming to be proposed to on bended knee. The tradition of kneeling to ask for someone's hand is a reminder of an era in which kneeling was a sign of deep respect, indicating an intense desire for something.