There is so called  "the pigeonhole principle"  in Math:


    If 7 pigeons are placed in 6 holes, then at least one hole contains 2 or more pigeons. 


In this joking form it is obvious and does not require more detailed proofs / explanations.




Further, in more general form, 


    if (n+1) pigeons are placed in "n" holes, then there is at least one hole containing 2 or more pigeons. 


In this form it is obvious, again, and does not require more detailed proofs / explanations.




Now let me formulate even more general THEOREM.


    If there are  n*m+1  items in  "m" containers, then there is at least one container containing  "n+1"  or more  items.



The proof is in three lines.


    If there is NO such a container, then the total number of items in "m" containers is NOT MORE than  n*m.


    It  CONTRADICTS  to the given part (!)


    This contradiction proves the statement.




Now let's return to our problem.


We have 731 = 2*365 + 1  students  (= items),  and

        365 days in the year (that are "containers" in this case).


From the Theorem, there is at least one container containing (2+1) = 3 or more items.

In other words, there is at least one day (one date in an year), when 3 or more students celebrate their birthdays.