。
The leaves on the trees start to fall in October.
Literal
Tree-[possessive-の]-leaves [topic-は] ten-month [at-に] fall-begin.
落ち始める is a compound verb built from 落ちる ('to fall') + 始める ('to begin'). The auxiliary 始める attaches to verb stems to mark the beginning of an action. Note that 始める (transitive) is used here even with an intransitive main verb, following a general rule in Japanese compound verbs where the aspectual auxiliary controls transitivity.