The left hand plays only the lowest line of notes in the bass clef—no chords. The right hand plays two independent voices simultaneously: the high one is the melody, and the lower one is a subordinated accompaniment. Chords result for the right hand where those two voices coincide; their span is generally within an octave, so they shouldn't be too difficult to manage.
If in your edition some of the right hand's lower voice is notated in the bass clef, that's done solely to avoid an excessive number of ledger lines between the staves.