。
Let me check what time the train arrives.
Literal
Train [topic-は] what-time [at-に] arrives [question-か] look-up-try-みよう.
Embedded question structure: 何時に着くか forms a subordinate question, then 調べる ('investigate, look up') takes it as content. 調べてみよう is ~てみる ('try doing') + volitional — 'let me try to look it up.' A natural self-directed or inclusive 'let me/let's check.'