I came across this article on Make Magazine on "The Brain Machine" which will supposedly put one into a trancelike meditative state via a combination of sound and light pulses.
Is there any scientific evidence supporting the effectiveness of such a device?
I have not heard about any similar devices.
You have to break down the question in two parts.
(1) Is it possible to entrain brain oscillations by presenting oscillating lights or tones?
Yes, it is possible to evoke "response phenomena" in the brain by presenting participants with oscillatory stimuli. The technical term for this phenomenon is "steady state evoked potential" and it was demonstrated for the visual (e.g., Herrmann, 2001) and as well as for the auditory domain (e.g., Zaehle et al., 2010).
(Please note that these references are only examples from one group. A more comprehensive overview can be found at PubMed.)
Thus, the devices you linked to might work in principle, that is, they might be able to entrain different brain oscillations.
(2) Is it possible to put someone into a trancelike meditative state by entraining certain brain oscillations?
It has been shown that the different frequency bands in which brain oscillations can be found are connected to perceptual and cognitive phenomena. (For a short introduction see Wikipedia on brain oscillations). Furthermore these frequency bands are not isolated, but the interweaving of these bands (i.e., cross-frequency coupling) is not fully understood yet (see for example Jensen & Colgin, 2007).
It has been shown, however, that the manipulation of brain oscillations can alter perception (see for example publications on phase resetting.)
In general, the topic of brain oscillations and its experimental manipulation is quite hot at the moment and receives a lot of attention, especially since new non-invasive methods for oscillatory brain stimulation are available (namely repetitive transcranial magnetic stimulation and transcranial alternating current stimulation). For a review, see Thut et al. (2011).
To come back to your initial question: no, I am not aware of any study connecting the entrainment of brain oscillations to trancelike meditative states. This, however, does of course not mean that there are no studies. Furthermore, that does not mean that you cannot reach meditative states while using this device. (Personally, I would think that you can reach meditative states with almost any stimulation - if you are good at mediation.)
References
Herrmann CS. (2001). Human EEG responses to 1-100 Hz flicker: Resonance phenomena in visual cortex and their potential correlation to cognitive phenomena. Experimental Brain Research, 137: 346-353.
https://doi.org/10.1007/s002210100682Jensen O, Colgin LL. (2007) Cross-frequency coupling between neuronal oscillations. Trends in Cognitive Science 11(7):267-9.
https://www.sciencedirect.com/science/article/abs/pii/S1364661307001271Thut G, Schyns GP & Gross J (2011) Entrainment of Perceptually Relevant Brain Oscillations by Non-Invasive Rhythmic Stimulation of the Human Brain. Frontiers in Psychology 2011; 2: 170.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3142861/Zähle T, Lenz D, Ohl FW, Herrmann CS. (2010). Resonance phenomena in the human auditory cortex: Individual resonance frequencies of the cerebral cortex determine electrophysiological responses. Experimental Brain Research, 203, 629-635.
https://doi.org/10.1007/s00221-010-2265-8
