When placed on the ground and powered, the FALCOR beacon will collect items on the ground within the current and any adjacent segments (use F9 to see segment boxes) and present advertise them to any FALCOR MK1 within 128 blocks horizontally.
It must be powered, but the power consumption is quite low. It can hold one item, and the item stays inside the FALCOR Beacon until a FALCOR MK1 picks it up. Thus, the transfer rate of items depends on the distance between FALCOR Beacon and FALCOR MK1.
When placed on top of a hopper, the FALCOR beacon does not require power and will pull an item from the hopper and advertise it to any Logistics FALCOR within 128 blocks horizontally. This is typically used to provide Cutter Heads to Ore Extractor.
In P18, a FALCOR beacon must have sky access in order to function. If any block is between it and the open sky, it will not function at all. In P19 you need at least 8m free space over the beacon to work underground. In P20.09+, you need at least 16m of free space over the beacon to work underground.