In plain words
Krishna, please dispel this doubt of mine completely. Other than You, no one is fit to remove this doubt.
Word for word
You ought to cut this doubt of mine completely, O Kṛṣṇa; for other than You a remover of this doubt is not found.
The commentators
Bhagavadgītābhāṣyam· bhāṣya
एतत् मे मम संशयं कृष्ण च्छेत्तुम् अपनेतुम् अर्हसि अशेषतः । त्वदन्यः त्वत्तः अन्यः ऋषिः देवो वा च्छेत्ता नाशयिता संशयस्य अस्य न हि यस्मात् उपपद्यते न सम्भवति । अतः त्वमेव च्छेत्तुमर्हसि इत्यर्थः ॥ ३९ ॥
In plain words
'This' — and 'me' here simply means 'mine': this doubt of mine, Kṛṣṇa, you are able to cut through, to lift away, leaving nothing of it behind. 'Other than you' means exactly that: other than you. No sage, no god, cuts this doubt down, destroys it. The 'for' means 'because' — because there is no such person; no one like that exists. So it is you alone who can cut it. That is the sense.
Close to the Sanskrit
'This' — 'me' means 'mine' — this doubt of mine, O Kṛṣṇa, you are able to cut, that is, to remove, without remainder. 'Other than you' means other than you: no sage or god is a cutter, a destroyer, of this doubt; 'hi' means 'because' — it is not possible, it does not occur. Therefore you alone are able to cut it — this is the meaning.