Last night, to try and find an answer to this question about how to change modes, first I re-read all 36 Amazon product reviews. Of those reviews, roughly 1/3 reported issues, either related to short time failure or inability to change modes, followed up by returns or trashing. Then I searched for similar products available elsewhere. Nothing else helpful to changing modes found either way.
Knowing little about RC systems. I decided to take a peek down the deep dark rabbit hole known as RC 3rd channel signal. What I learned after 90 minutes of various searches on RC message boards and YouTube videos is that details about RC 3rd (and 4th) Channel signal makeup is scarce.
Two useful tidbits I did find are this: 1) The signal transmitted (over the yellow wire) is a low frequency PWM (Pulse Width Modulated) 5V waveform. 2) In one video lacking much useful electrical detail, someone using a multi-channel controller connected to a different type of 5-mode light gizmo, got the gizmo to change modes by quickly double tapping the momentary switch assigned to that channel. I estimate his time between taps at between 1/8 to 1/4 second.
So as I see it, if no one else has any better/more informed ideas, options are this: return these flashing lights or try a couple of experiments on them. In both experiments, connect the light's Red wire to 5 VDC Positive and its Brown wire to the DC source's Ground.
First experiment, try quickly double tapping the Yellow wire in rapid succession (1/4 second apart) to the DC source Ground and waiting a moment to see if the flash mode changes.
If that doesn't advance to the next mode, second try the same procedure except double tapping the Yellow wire to the DC + terminal.
No guarantees that the second mode won't damage something, but I'm guessing that a couple of brief (5ms) connections to +5VDC won't hurt anything here.
Good luck and please let us know if either of these experiments are successful. 
