Oh great owl of Bohemia we besiege thee!, grant us thy council?

The word is "beseech", meaning beg, not "beseige." It's just standard quasi-religious stuff, imploring a deity (in this case an owl) to to answer our requests. It sounds as though they make a little statue of the owl and burn it as an offering, then read the pattern of the fire or the ashes as a message from the owl.