We raised both beef and dairy cows, but never heard this word

I suspect that you always had enough grain/food/grass for your cattle then. I've heard this term used a fair bit - particularly when I was growing up in the country. People who lived in town on small blocks of land would often have the horses that their kids had as pets agisted on properties outside of town. My uncle who has a farm in one of the more arid regions of NSW occasionally agists his animals onto the property of a neighbour, who doesn't have animals, and pays a fee to do this (often in kind, rather than specific monetary amount - the barter system still operates in the country to a degree).

In Oz, the term 'agistment' implies more than just feeding, it includes the pasturing as well. On checking my facts, I find that the Macquarie Australian Dictionary says:

agist
// verb (t) 1. to take in and feed or pasture (livestock) for payment. 2. to lay a public burden, as a tax, on (land or its owner). [Middle English, from Old French à to + giste resting-place]
--agistment, noun
--agistor, noun

Hmmm... interesting!