The key verb here is なる, which basically means "to become". In its simplest use case, you can apply it to adjectives:
- 強くなる
to become strong
- 強くなくなる
to become not strong → to stop being strong / to become weaker
なる is more versatile than English "become"; you can apply it also to verbs and express a change of situation (ability, habit, etc.). However, you need ように between the two verbs.
- 学校に行くようになる
to start to go to school
- 自転車に乗れるようになる
to come to be able to ride a bicycle
You can construct a negative version of these:
- 学校に行かないようになる
to start not to go to school → to stop going to school
- 自転車に乗れないようになる
to come to be unable to ride a bicycle
Now, remember that Japanese ない conjugates like an adjective. This means you can directly attach ない and なる without ように, just like the first example above!
- 学校に行かなくなる
to start not to go to school → to stop going to school
- 自転車に乗れなくなる
to come to be unable to ride a bicycle
You can use this handy shortcut only when the main verb is negated, but it's usually preferred over ~ないようになる. Therefore, 熊が近くに来なくなります is the same as 熊が近くに来ないようになります, and means "bears will stop coming closer" or "(with this,) bears won't come nearby anymore". Notice the change in state is expressed by なります, which is the key difference from 熊が近くに来ません. (By the way, to simply negate a verb, say 来ません instead of 来ないです.)
For details and more examples, see this article (especially the chart): apieceofsushi.com: JLPT N4 Grammar 文法 -you ni naru 〜ようになる ‘to come to point that’ ‘to start to’ ‘to become able to’