The trees in my yard are starting to put out leaves.
Literal
I [of-の] garden [of-の] tree [subject-が] leaves [object-を] put-out begin-is.
~はじめる is an aspectual auxiliary attached to verb stems meaning 'to start [verb]ing.' Used freely with any volitional verb. 葉を出す ('to put out leaves') is the standard botanical expression for budding. A seasonal observation typical of spring.