This kind of conflicting evidence regarding the effectiveness of binaural beats to produce valid and reliable changes in brain waves abounds in the literature. For example, Rosenfeld, Reinhart, & Srivastava, (1997) found that in a sample of normal college students, alpha and beta audiovisual stimulation showed evidence of brainwave entrainment, but baseline levels of alpha and beta among the participants affected the observed degree of entrainment, producing significant individual differences in response. López-Caballero & Escera (2017) found that administration of binaural beats in the various frequency bands produced no changes in EEG spectral power between the time periods of baseline and those periods with beats presented. Likewise, Wahbeh, Calabrese, Zwickey, & Zajdel (2007) found no effect on brainwaves with the administration of alpha frequency binaural beats. It is easy, however, to find personal testimonials online. 
Most of these websites give some brief explanation of entrainment. The example you hear most often is that of Dutch polymath Christiaan Huygens, who in 1665, hung two pendulum clocks next to each other on a wall. He noticed that the pendulums eventually matched each others' frequency, but always in antiphase, opposite to each other, as if canceling each other out. He'd try disturbing one or setting them in sync, but they'd always return to the same antiphase synchronization. Huygen's experience is widely touted on binaural beat websites as a demonstration of how systems can become spiritually connected through some energy field. However, they misunderstand what happened, and have not read the full story. Huygens also tried taking one clock off the wall, and as soon as they were no longer physically connected to one another via the actual wall, the effect disappeared. It was not the proximity of the clocks to one another that created the entrainment; it was their physical, mechanical connection to one another. As each pendulum swung it imparted an infinitesimal equal and opposite reaction to the wall itself. With two clocks on the wall, the system naturally sought the lowest energy level, according to the laws of thermodynamics; and both pendulums would thus swing exactly counter to each other, minimizing the system's total energy.
Transparent Corp's Research Area is arguably the most comprehensive resource for collated brainwave entrainment research.   Update: the main research area on Transparent Corp's website is currently being updated, so it is offline.  However, you can still access their peer-reviewed research paper as a PDF here: “A Comprehensive Review of the Psychological Effects of Brainwave Entrainment“.
Brainwave entrainment also happens with the use of pulsating light, and visual and auditory stimuli are sometimes combined for additional effect and visual stimuli is used alone. Using brainwave entrainment techniques is safe for almost everyone, the exception being pregnant women and people who have seizure disorders who should check with their physician before using these methods.
Theta brainwaves occur most often in sleep but are also dominant in deep meditation. Theta is our gateway to learning, memory, and intuition. In theta, our senses are withdrawn from the external world and focused on signals originating from within. It is that twilight state which we normally only experience fleetingly as we wake or drift off to sleep. In theta we are in a dream; vivid imagery, intuition and information beyond our normal conscious awareness. It’s where we hold our ‘stuff’, our fears, troubled history, and nightmares.
Brainwave entrainment is a method to alter the frequency of your brainwaves by listening to audible tones containing the desired frequency. So if you want to sleep, you would listen to audio that begins at your current frequency and slowly descends to a delta frequency. By introducing these new lower frequencies, your mind begins to lower its own brainwaves to match this new frequency.

A more extensive study of over 100 participants who were undergoing general anesthesia for a day procedure, reported a decrease in pre-operative anxiety. The participants in this study listened to 30 minutes of binaural beats before surgery, but the researchers noted that people experiencing high levels of pre-operative anxiety could listen to binaural beats for up to 1 hour before anesthesia to reduce levels of anxiety.