As was the case for calculating the number of the Beast (Rev. Rev. 13:18+), wisdom is required to understand the next portion of the mystery revealed by the angel.
the seven heads are seven mountains
As Woods observes, if the seven mountains are to be taken as the seven hills of Rome, then it is difficult to see why special wisdom is said to be required in order to understand the revelation provided by the angel:
It seems odd that the seven hills should be equated with the well-known topography of Rome because Revelation Rev. 17:1+: indicates that the identification of the hills calls for special wisdom. Why should such a well-known geographical locale to Johns first century audience require special theological and symbolic insight for proper identification?1As we shall see in the next verse, the seven heads are seven kings. Here they are said to be seven mountains. The relationship between kings and mountains is well-established in Scripturemountains represent the power of kingdoms and their individual kings (Jer. Jer. 51:25; Dan. Dan. 2:35; Zec. Zec. 4:7). These seven mountains, together with the eighth (Rev. Rev. 17:11+), will eventually be destroyed by the stone which is Christ, the Messianic King (Rev. Rev. 20:4+). His kingdom is destined to become a great mountain and fill the whole earth (Dan. Dan. 2:35).
This symbolic understanding of the seven mountains seems buttressed by the fact that the harlot sits on or beside seven mountains (Rev. Rev. 17:9+) just as she sits on or beside the many waters (Rev. Rev. 17:1+). Since the waters are symbolic of peoples, multitudes, nations, and tongues (Rev. Rev. 17:15+), consistency seems to dictate that the seven mountains are symbolic as well.2See #4 - Seven Heads/Kings.
on which the woman sits
That which she sits upon supports her and she controls it. It is difficult to know which side of this symbiotic relationship is more important, although Scripture seems to indicate her corrupting influence is what God especially opposes. Some interpret her sitting as denoting her location. That she is located upon seven hills which are then said to be the hills of Rome. But this is not the predominant meaning of her sitting which speaks more of support and control than locale:
The reference to the seven mountains (Rev. Rev. 17:9+) which are seven heads (Rev. Rev. 17:8+) actually belong to the beast (Rev. Rev. 17:3+, Rev. 17:7+; Rev. 13:1+) and not the woman named Babylon. Thus, these seven heads or mountains really have nothing to do with the entity Babylon at all. It is possible to argue that the woman is still associated with the seven hills because she is sitting on them. However, it is better to see this as referring to the womans control rather than her location. The other references to the woman sitting also refer to her control. Revelation Rev. 17:1+ portrays the woman sitting on many waters. Revelation Rev. 17:15+ explains that the waters represent peoples, multitudes, nations, and tongues. Thus, Revelation Rev. 17:1+, Rev. 17:15+ show the harlots control over the entire world. Furthermore, Revelation Rev. 17:3+ depicts the woman as sitting on the beast, which again indicates control rather than location. Thus, if the harlots sitting indicates control rather than location twice in Revelation Rev. 17:1+, then consistency would seem to dictate that the harlot sitting on the seven hills in Revelation Rev. 17:9+ would also indicate control rather than location.3See Babylon is Rome?.
1 Andy Woods, What is the Identity of Babylon In Revelation 17-18?.