Consider the integer a. There are three possibilities.
Case 1: There is an integer k such that a=3k
Case 2: There is an integer k such that a=3k+1
Case 3: There is an integer k such that a=3k+2
In case 1, a is divisible by 3.
In case 2, a+2=(3k+1)+2=3k+3=3(k+1) so that a+2 is divisible by 3
In case 3, a+1= (3k+2)+1=3k+3=3(k+1) so that a+1 is divisible by 3
Notice that all I am using in this argument is the definition of divisibility by 3. If you understand he definition, then the argument is straightforward and almost obvious.