I just looked at the Wikipedia article on beamforming and i can understand your confusion. It ‘s trying to explain too much at once. There are some nomenclature issues as well, MUSIC is a direction finding technique. It doesn’t form beams in the literal sense.
Unfortunately a lot of articles as well as books are written by people who understand the material for other people who understand the material.
You should also be aware that beamforming isn’t always or even practical solution for many audio situations. Shotgun microphones or parabolic reflectors are often good and potentially cheaper solutions.
The simplest way to beamform is to place all your microphones on a plane that is perpendicular to the direction of the source you want to amplify and then sum the outputs of all the microphones. This is based on the assumption that the source is far enough away so that you can neglect the curvature of the wavefront. This assumption depends on the assumption that the source can be considered a point emitter.
Another way to put this is that the acoustic delay time from the source to each microphone is the same. The Wikipedia article used the term constructive interference to say the same thing.
If locating the microphones on the coherent wavefront is not possible, one introduces delays in each microphone so that each summed microphone has the same acoustic distance travel time from the source when coherently summed. This introduction of delays can be very general and permit the array to coherently sum the sounds from many directions.
Each microphone should have a clear line-of-sight to your source. Microphones typically have a directional response as well so it helps to point each microphone at the source as well. One cannot compensate for the individual directivity of each microphone by introducing delays. Actually a lot of the material written about arrays makes the unrealistic assumption that each microphone has an omnidirectional response.
There are numerous effects that complicate conventional delay-and-sum beamforming as i have described it. The source sounds can arrive at the array from more than one direction which is called multi-path. The signal can cause destructive interference with itself in this circumstance.
You can have another loud source that impinges on the array, so even if not coherently summed, still interfere with the source of interest. The way to treat this is to steer a second beam towards that source and judiciously subtract its sound from the sound of interest. There is an entire literature on optimal beamforming that treats this situation.
Summarizing, beamforming may or may not be worth it. It takes specialized hardware like synchronized multichannel analog-to-digital converters.
It isn’t cheap compared with a shotgun microphone. Beamforming is often done because nothing else is possible.