'To beat the air' means to keep doing something despite its apparent pointlessness or futility. Option A is the correct answer because 'to make vain efforts' means to attempt something which is of no value, unsuccessful or useless; failing to achieve a purpose.
The other options are incorrect because 'to brag and boast' means to speak too proudly about what you have done or what you own, 'to give airs' means to act in a way that shows one thinks one is better than other people, 'to submit' means accept or yield to a superior force or to the authority or will of another person and 'to acknowledge defeat' means to admit and accept failures or losses.