Lawn boots may refer types of human footwear, but that's not what I'm writing about. Instead, let's talk horses.
Back in the 1800s, horses were used to pull lawn rollers and lawn mowers. Horse-drawn lawn mowers were a considerable improvement over scything a lawn, or, alternatively, allowing grazing animals to keep the grass short. They were sold from about 1842 to 1942, at which time gasoline motors and smaller machines allowed push mowers and riding lawn mowers to take over.
However, having a horse walk over your lawn did not necessarily improve its aesthetic appeal, as they could raise divots and churn up muddy spots. The original solution was to have the horse-pulled lawn rollers flatten these out, or to use unshod horses wearing light woolen mitts. But as technology moved forward, the preferred solution became buckle-on overshoes, commonly known as lawn boots.
These shoes were made out of heavy leather with lightly studded soles, and were designed to be easily fit over the horseshoes, and removed quickly when mowing was done. The fitting of lawn boots was a matter of some concern, although it appears that eventually a semi-standardized system of five sizes, ranging from pony through donkey through horses, was established, with variants for 'straight' and 'slanted' hooves.
All of which went down the drain when motorized mowers took over. A related product, hoof boots are still on the market for trail-riding with 'barefoot' (unshod) horses, although these look more like sneakers than the traditional leather lawn boots, with treaded leather soles and nylon uppers with Velcro fasteners ('socks', AKA pastern wraps are also used if the shoes rub). These modern shoes are not designed to protect the lawn, although they probably would do so, but to protect the horses' foot. It would not be kind to the horse to replace them with an antique lawn boot; they are completely different beasts.