The phrase 'carry on' means 'to continue or proceed'; it is usually used in imperative sentences. For example: You can carry on with your work.
'Continue' means 'persist in an activity', consider the sentence: You can continue with your work. Hence option D gives the meaning of the phrase and D is correct.
'Start' means 'to begin', but 'carry on' is used in the middle of an action, that action has already started. Hence A is incorrect.
'Execute' means 'to put into effect', for example: 'to execute a plan'. It does not mean the same as the given phrase. Hence option B is incorrect.
'Finish' means 'to bring an end to something' whereas 'carry on' means 'to continue and not end'. Hence option C is incorrect.