I recently came across an article about the Murchison Wide Field Array which said directivity is achieved by Electronic Beam Steering and not by Mechanical methods. It would be helpful if someone can tell me: how do they achieve it?
This can actually mean a couple of different things, but in the case of steering an MWA tile, Jeremy is quite correct. Each tile has a 'beamformer' box that all of the individual antenna cables plug into. This box physically delays each signal appropriately (by having a longer path to go through for more delay) based on the signals sent to it for each pointing. Because of the quantised path lengths there are what are called 'sweet spots' that are used, in other words arbitrary pointings don't normally happen you just dial in the closest 'sweet spot' to where you'd like to point.
There are also version that do this in software (and in fact in a special phased mode the MWA does this between tiles sometimes), where 'turns of phase' are used instead but it is essentially the same idea.