、・・・。
It seems I'd been dozing off...
Literal
Apparently, I [topic-は] was-dozing [seems-らしい]...
微睡む (まどろむ) is a literary-leaning verb for 'to doze / drift off into a light sleep,' more evocative than 居眠りする or うとうとする. どうやら 'apparently / it seems' frames the statement as a reluctant or belated realization. ~らしい 'seems / apparently' compounds the inferential feel — the speaker is piecing together evidence about their own recent state. ~ていた marks the ongoing past state of dozing.