Well, a vector is a type of stack, lets say a stack of pancakes since we all love pancakes. When you push back, you basically add a pancake but you are adding it from the back of the stack. There is also push front which add to the other end. Pop back and pop front will remove from the back and front respectively.
So lets say you have a vector of integers where numbers[0] = 5 and numbers[1] = 2. Now you do a push back with an integer of 3, your stack will now look like numbers[0] = 3, numbers[1] = 5, and numbers [2] = 2. See how they all moved. Actually, I think I actually demonstrated a push front, a push back just adds it to the end.
You can't just declare it on the end of a vector because a vector is sized according to what is in it. So if you only have 2 integers pushed into a vector, then it will only hold 2. It will grow as you need it and shrink as you don't. You can manipulated the data already in there, but you can'y just add any without first pushing.
I don't think I explained it well, if not I can try again in another way. The way I learned is that I looped the output so that I saw the change each time I pushed front or back (deque will show it better since they handle all the things vectors handle plus push and pop fronts).