The OSEPP Mega 2560 board requires between 100 and 400 mA of current for normal operation, not including any LEDs or other peripherals that are on and need power. Presumably the R3 Plus board needs as much, or more.
This power requirement hits the higher end when the USB-to-serial IC (the one near the USB socket) is transmitting / receiving, or attempting to.
When operated from the USB port, the board has up to 500 mA available to it from the computer it is connected to, so power is not a problem.
The OSEPP solar cell generates its full capacity of 7.2 V at 200 mA only under bright daylight. When used indoors, especially with normal room level artificial lighting, the voltage may fall below the 6 Volts minimum required by the Mega 2560 board.
The problem described appears to be of insufficient power delivered by the solar cells to the board. The USB TX/RX lights blink because the bootloader attempts to check for any communication from the Arduino IDE, and this will drive up the board's current draw. In this situation, the solar cell simply cannot supply sufficient power, and the board resets due to a brown-out i.e. insufficient voltage / current reaching the voltage regulators on board.
Power the board by a 9 Volt battery to check this hypothesis. If it behaves normally under battery power, then you may need to connect two or more identical solar panels in parallel to supply sufficient power. This is described in the solar panel datasheet linked above.