。
She couldn't properly explain why she was late.
Literal
She [topic-は] lateness [possessive-の] reason [object-を] well explain could-not.
うまく ('well / skillfully') modifies 説明できなかった — she couldn't explain effectively. 遅刻の理由 ('reason for being late') chains two nouns with の. 説明できなかった is the potential negative past of 説明する. The sentence suggests she had a reason but failed to articulate it convincingly.