The correct sentence is (B) He doesn't like to eat vegetables. This sentence correctly uses the auxiliary verb 'does' with the third-person singular subject 'He' in the negative present simple tense. Following 'doesn't', the main verb 'like' must be in its base form.
Option (A) 'He like to eat vegetables' is incorrect because for a third-person singular subject ('He'), the verb in the present simple affirmative should be 'likes' (subject-verb agreement). Option (C) 'He doesn't liking to eat vegetables' is grammatically incorrect; after 'doesn't', the verb must be the base form 'like', not the gerund/present participle 'liking'. Understanding subject-verb agreement and the correct structure for negative sentences in the present simple is crucial here.