In plain words
Therefore, with the sword of jñāna cut this doubt born of ignorance that dwells in your heart. Take refuge in yoga and arise, Bharata.
Word for word
Therefore, having cut with the sword of knowledge this doubt of your own, born of ignorance, lodged in the heart, resort to yoga; stand up, O Bhārata.
The commentators
Bhagavadgītābhāṣyam
यस्मात् कर्मयोगानुष्ठानात् अशुद्धिक्षयहेतुकज्ञानसञ्छिन्नसंशयः न निबध्यते कर्मभिः ज्ञानाग्निदग्धकर्मत्वादेव, यस्माच्च ज्ञानकर्मानुष्ठानविषये संशयवान् विनश्यति —
तस्मात् पापिष्ठम् अज्ञानसम्भूतम् अज्ञानात् अविवेकात् जातं हृत्स्थं हृदि बुद्धौ स्थितं ज्ञानासिना शोकमोहादिदोषहरं सम्यग्दर्शनं ज्ञानं तदेव असिः खङ्गः तेन ज्ञानासिना आत्मनः स्वस्य, आत्मविषयत्वात् संशयस्य । न हि परस्य संशयः परेण च्छेत्तव्यतां प्राप्तः, येन स्वस्येति विशेष्येत । अतः आत्मविषयोऽपि स्वस्यैव भवति । छित्त्वा एनं संशयं स्वविनाशहेतुभूतम् , योगं सम्यग्दर्शनोपायं कर्मानुष्ठानम् आतिष्ठ कुर्वित्यर्थः । उत्तिष्ठ च इदानीं युद्धाय भारत इति ॥ ४२ ॥
In plain words
So: because practising karma-yoga wears away the mind's impurity, and out of that clearing knowledge rises and cuts doubt clean through, such a person is no longer bound by anything he does — his actions have been burnt in the fire of knowledge, and that alone is the reason. And because the man who keeps doubting, unsure whether to take up knowledge or action, is simply ruined —
So this doubt — the most sinful thing there is, born of ignorance, arising from a failure to discern — sits lodged in the heart, in the intellect within the heart. 'With the sword of knowledge': that right vision which cuts away the taint of grief and delusion is itself the sword, the blade. With that sword of knowledge, cut your own doubt — because doubt, by its very nature, is always about the Self, and never belongs to anyone else. No one's doubt can ever be cut by someone else in a way that could be called cutting 'one's own' doubt; that just isn't how it works. So even though doubt concerns the Self, it is always, strictly, one's own to deal with. Having cut away this doubt, which is nothing but the cause of your own ruin, 'take up yoga' — the means to right vision, the actual doing of your work — meaning: do it. That is the meaning. 'And rise now, for battle, Bhārata' — so Krishna says.
Close to the Sanskrit
Since, from the performance of karma-yoga, he whose doubt has been severed by the knowledge that is caused by the destruction of impurity is not bound by actions — precisely because of the state of his actions being burnt in the fire of knowledge — and since one who has doubt with regard to the performance of knowledge and of action perishes —
Therefore, the most sinful, born of ignorance — arisen from ignorance, from non-discrimination — 'situated in the heart', situated in the intellect within the heart, 'with the sword of knowledge' — right vision, which removes the taint of grief, delusion, and the like, that is, knowledge — that very knowledge is the sword, the blade — with that sword of knowledge, one's own doubt, since doubt has the Self as its very object. For never is another's doubt fit to be cut by someone else, such that it could be qualified as 'one's own'. Hence, even being about the Self, doubt belongs to oneself alone. Having cut this doubt, which is the very cause of one's own ruin, 'take up yoga' — the means to right vision, the performance of action — meaning, do it. This is the meaning. 'And rise up now for battle, O Bhārata' — so it is said.