Having read the paper...
The LEDs are only used for the downlink. (A 140W laser is used for the uplink.) They transmit using 1000 lumens worth of LEDs with a peak power of 140W and use lenses to focus the beam. It is reckoned it will emit a 19.9 dBW signal -- that's visual magnitude of 1, so naked eye visible under ideal conditions.
Those fins are a conventional aluminium heat sink. But the LEDs only transmit for 2min/orbit and then they let the heat dissipate. Radiation benefits from surface area every bit as much as convection, so perhaps.
The signal is received using a commercial 30cm telescope. Accounting for losses that's a -107.3 dBW signal giving a speed of 8.85E4 bit/s or 700 photons per bit. But they plan to use "deep learning" on both ends to develop a modulation scheme -- they haven't done so,.
They don't say much about the uplink to solar panels, referring to other work. But it uses a laser mounted on the receiving telescope.