Try to take smaller numbers to make it clear.
Take 28 and 10.
The even integers are: 10, 12, 14, 16, 18, 20, 22, 24, 26, 28
If you add them all together you'll get: 190
You get the average when adding 28+10)/2
So 19 is exactly between 10 and 28, right.
If you add 10 and 28 you make the range from 0 to 38 and division by two results in 19 - be careful it's not the same as 28-10
Now you need the no. of the terms: 28-10=18 (but also with odds) therefore divide by 2: 18:2=9 (now without odds) but as it says inclusive add 1: 9+1=10
Now the sum off all even integers between 10 and 28 is: 10*19=190 which is the same
Hope this helps