In plain words
Out of compassion for them alone, dwelling in their inner being, I destroy the darkness born of ajñāna with the radiant lamp of jñāna.
Word for word
For them alone, out of compassion, I, abiding in their inner being, destroy the darkness born of ajñāna with the radiant lamp of jñāna.
The commentators
Bhagavadgītābhāṣyam
किमर्थम् , कस्य वा, त्वत्प्राप्तिप्रतिबन्धहेतोः नाशकं बुद्धियोगं तेषां त्वद्भक्तानां ददासि इत्यपेक्षायामाह —
तेषामेव कथं नु नाम श्रेयः स्यात् इति अनुकम्पार्थं दयाहेतोः अहम् अज्ञानजम् अविवेकतः जातं मिथ्याप्रत्ययलक्षणं मोहान्धकारं तमः नाशयामि, आत्मभावस्थः आत्मनः भावः अन्तःकरणाशयः तस्मिन्नेव स्थितः सन् ज्ञानदीपेन विवेकप्रत्ययरूपेण भक्तिप्रसादस्नेहाभिषिक्तेन मद्भावनाभिनिवेशवातेरितेन ब्रह्मचर्यादिसाधनसंस्कारवत्प्रज्ञावर्तिना विरक्तान्तःकरणाधारेण विषयव्यावृत्तचित्तरागद्वेषाकलुषितनिवातापवरकस्थेन नित्यप्रवृत्तैकाग्र्यध्यानजनितसम्यग्दर्शनभास्वता ज्ञानदीपेनेत्यर्थः ॥ ११ ॥
In plain words
For what purpose, and for whose sake, do you give your devotees that buddhi-yoga which destroys the very cause of what blocks the way to you? Expecting the question, he says:
Thinking, 'how might these very people come to their highest good' — out of sheer compassion, out of pity — I destroy that darkness: the darkness born of ignorance, arisen from a failure to discriminate, made of nothing but false notions, the darkness of delusion. I do this abiding in the very seat of the Self, in the innermost dwelling of the mind itself, with the lamp of knowledge — a lamp whose flame is the very sense of discrimination, its oil the grace and affection devotion pours into it, fanned bright by the wind of absorption in thinking of me, its wick the understanding refined by disciplines like celibacy, resting on the steady base of a mind grown dispassionate, burning in a windless inner room undisturbed by any pull of attachment or aversion because it has turned away from the objects of the senses — and blazing with the clear vision born of constant, one-pointed meditation. That is what is meant by 'the lamp of knowledge.'
Close to the Sanskrit
For what purpose, or for whose sake, do you give to them, your devotees, the buddhi-yoga that destroys the cause of the obstruction to attaining you? — such a question being expected, he says:
Thinking 'how indeed might their welfare come to be' — for the sake of compassion, out of pity — I destroy that darkness born of ignorance, arisen from lack of discrimination, characterized by false notions, the darkness of delusion, that very darkness — I, abiding in the Self's own state, that state being the seat of the inner organ, abiding precisely there — with the lamp of knowledge, having the form of the notion of discrimination, anointed with the oil of the grace and affection of devotion, fanned by the wind of absorption in contemplation of me, whose wick is the understanding possessed of the refinement of disciplines beginning with celibacy, resting on the base of a dispassionate inner organ, situated in a windless chamber unclouded by the attachment and aversion of a mind withdrawn from sense-objects, radiant with the right vision produced by constant, sustained, one-pointed meditation — this is what is meant by 'the lamp of knowledge.'