1. To surround with hostile forces
2. To harass, distress, or to present a request
3. To crowd around; hem in.
The word ‘besiege’ can be thought of as ‘be+seize’. When surrounded by hostile forces, you are most likely to end up getting up seized by them and hence this can be related to the meaning of the word ‘besiege’.
1. The king’s army sought to besiege the castle before nightfall.
2. Reporters would besiege the winner of the contest for interviews until he could stand it no longer and so he finally gave in.
3. The city counselors sought to besiege the mayor to repair the roads before winter.