They are very popular in Asia and Europe and have been being used for decades. Trailers are presumed to be more stable. Trailers with rear triangle hitches have the advantage of affecting bike handling much less than bike seats, or even trailers with seatpost hitches. Trailers also have the advantage of being able to carry children to a larger size and multiple children. Kids can be set-in the trailer with toys, books, drink, food and other amusements and enjoy relatively long tours.
Trailers can be more weather proof.
But the child sit low with a restricted view and the trailer has a larger foot-print so they may be a problem on narrow roads and in congested urban areas. Rather than attracting harassment trailers seem to more often generate curiosity and rubber-neckers. Face forward designs allow easier communication between the cyclist and the child. The child is also easy to monitored with the use of a handlebar bar-end mounted rear view mirror. Taking an infant on a bike in a backpack potential more dangerous.