I looked up the quote and found "A History of Britain Ep. 3 - Dynasty". In the sentence just before you posted, they mention troubadours and courtly love.
1) The troubadours sang songs that dealt with chivalry and "courtly love". For some reason, this caused many women of nobility to desire, fantasize, and often have sex with the troubadour.
2) In the beginning, around the time of Eleanor, the troubadours were usually men from nobility, and often they were classified as knights (some called themselves "poor knights"). The sentence after "lovelorn lyrics of..." describes the knights/troubadours in general, and doesn't say these characteristics are directed at Eleanor. So, they were enslaved by beauties, means that many beautiful noble women demanded their services. No mention if Eleanor did.
The term "bent on" means they were 'determined to' (in this case besiege virtues).
3) The phrase, "besieging their virtue", again, applies to the noble women who are enamored with the troubadours. "Besiege" doesn't just mean "surround". It is to surround for a very specific purpose: to surround . . . in order to capture it or force its surrender. So, in this passage, "besieging their virtue" means the troubadours are putting pressure on the women to surrender their virtue (i.e., have sex with them).