Here is a definition ( with supporting quotes) from the OED which will help you understand the term, I think:

A wood on the side of a steep hill or bank: cf. HANGING ppl. a. 2b.

1789 G. WHITE Selborne lxxxvii, A considerable part of the great woody hanger at Hawkley was torn from its place and fell down, leaving a high freestone cliff naked and bare.

1822 in Cobbett Rur. Rides (1885) I. 179 These hangers are woods on the sides of very steep hills.

Of course what JA was doing when she was talking about the imagined grounds of Pemberely( and Allenham) was making a reference or allusion to the works of the Reverend William Gilpin who wrote about the visual appreciation of landscape in his works such as that shown below:

We know from Henry Austen’s biographical notice of his sister that JA was “enamoured” of Gilpin and his works, and it is clear from reading her texts that she was influenced by him( and was not above ridiculing his uniquely pompous style!).

Gilpin certainly admired the hanging woods that he observed in Derbyshire in his Observations etc of Cumberland and Westmoreland:

From Dove-dale we proceeded to Ilam; which is another very characteristic scene.

Ilam stands on a hill, which slopes gently in front; but is abrupt, and broken behind, where it is garnished with rock and hanging wood.

See page 229, Volume 2,Observations on Several Parts of England particularly the Mountains and Lake of Cumberland and Westmoreland relative Chiefly to Picturesque Beauty made in the Year 1772 by William Gilpin

I daresay , for there is no evidence that JA ever visited Derbyshire, that by using the device of hanging woods at Pemberley in Derbyshire JA felt safe to do so when she knew from her reading of Gilpin that such woods existed in the general area of her imagined great estate.