According to this primary school's website, heuristic play is defined as the following:
Heuristic play actively encourages exploration by using and developing children's senses. Children instinctively investigate objects that interest them, making discoveries through taste, touch, smell, sound and how they look. During the activity children explore different materials and objects without adult interference. The role of the adult is to support the children, collect objects, set out the activity and to observe.
Another site defines heuristic play based on the seminal work that gave this philosophy its name:
In their book, People Under Three, Elinor Goldschmied and Sonia Jackson coined the term heuristic play, to explain how to provide a more structured opportunity for this kind of activity. Heuristic play ‘consists of offering a group of children, for a defined period of time in a controlled environment, a large number of different kinds of objects and receptacles with which they play freely without adult intervention’. It is particularly useful for children in their second year who often seem unwilling to engage in any activity for more than a few minutes. According to the Oxford Dictionary, ‘heuristic’ means helping to find out or discover; proceeding by trial and error. It stems from the same root as Eureka – ‘I found it!’ Clare Crowther of Bridgwater College describes heuristic play as ‘an activity we use with one-year-olds, two-year-olds, and young threes, giving them the opportunity to experiment spontaneously with a wide range of non-commercial objects. Whilst the heuristic play session is in process, adults need to remain seated and quiet. This supports children in making their own choices and discoveries.’
Finally, this early childhood expert wrote an article that BabyCentre.co.uk linked to. Actually, BabyCentre.co.uk has a ton of results from parents (read: moms) who've tried this and really enjoyed it, as well as felt their kids love it a lot.
As for why it's so popular: probably a combination of two things.
Engaging in heuristic play is fairly inexpensive. Basically, you use household items in creative ways. Given the last five or so years' economic performances, a back-to-basics approach is unsurprising. Parents also have more control over the "toys," as they can have natural wood or plastics that are food-grade rather than potentially sketchy toys made in countries with less stringent health and safety guidelines for children's clothes/toys or the paint/sealants that go on said objects.
This is probably the more important: parents love watching their children discover, which is inherently what heuristic play / treasure baskets encourage. No parent I know doesn't completely light up when they tell stories about their babies first stacking blocks or using pots and pans and stirring spoons for an impromptu drumming jam session. It appeals to our (read: adults') own sense of wonderment that their kids have such wonderment of their own. But that's just my humble opinion.