밑동 is the bottom part of the tree

밑동만 is composed of
밑동 + 만
만 is a particle which means "only"

밑동만 남겨 놓다 means "leave only the bottom part of the tree"

밑동이 is composed of
밑동 + 이
이 is a subject particle which means "be"
밑동이 최고야 means "the bottom part of the tree is the best"

최고야 is composed of
최고이(다) + 아
최고이다 means "the best"
아 is an ending particle.

찾아왔지요 is composed of
찾아오(다) + 았 + 지요.
찾아오다 means "come to see something"
았 is past tense particle
지요 is an ending particle

나무를 찾아왔지요 means "He came to see the tree".

거라고는 is composed of
거 + 라고 + 는
거 means "thing"
라고 means "called"
는 is a topic marker
거라고는 means "something meaningful to do"

나는 너에게 줄 거라고는 이제 아무것도 없어 means
Now I have nothing meaningful to give you.

낮잠도 is composed of
낮잠 + 도
낮잠 means "nap"
도 is a particle which means "also"

낮잠도 잤지요 means
He had a nap also.

실버시타인 is a name Silverstein

술래잡기도 is composed of
술래잡기 + 도
술래잡기 means "hide-and-seek"
도 is a particle which means "also"

술래잡기도 하다 means
He also played hide-and-seek.

따 is composed of
따(다) + 아
따다 means "pick"
아 is a particle which means "and"

열매를 따 먹기도 하였어요 means
He also picked and ate the fruit.