When adding a number to 10, you keep your ones place number (the number to the farthest right) and raise the tens place number (the number to the left of the ones) up by 1.
In 9+10, you keep the 9 since its in the ones place.
Since there is only a 9 and no other number next to it, the 1 in 10 is added onto the 9 to form 19.
You can think of it like there is an invisible 0 in front of 9 to hold the tens place (09) so that's why we add on the 1 to that invisible 0 to get 19.
Here is how it would look stacking the numbers to add them: