According to Merriam Webster Dictionary, battle-tested means,
Having experienced and been toughened by battle or shown to be reliable and effective by being used in war.
Suppose we equate traditional wars to software development, where the warriors are the developers, and the enemies are bugs. In that case, a battle-tested library is a library that has little to no bugs and can be used confidently in production. A good example of such a library is JQuery.
JQuery has been in existence since 2006. Most of the bugs within it have already been fixed, and it has confidently been used in projects of all sizes.
The most important factor here is time. This doesn’t mean that all libraries created in 2006 are battle-tested as the library must have a significant number of users. Here are a few libraries that fit into this definition: