We can order the elements of the sets A and B in ascending order that is :
$A$={$a_1,a_2,a_3,...,a_|A_| $} , $B$={$b_1,b_2,b_3,...,b_|B_|$}
Now , firstly we add $a_1$ with all elements of $B$, so we get $|B|$ elements.
Now if we try to do so, there can be several repetitions of the same value, hence we cannot correctly estimate the bound of set $A+B$.
$\therefore$ what we do is add the remaining elements of $A$ i.e $a_2,a_3....,a_n$ with $b_|B_|$ , this will surely generate a new value every time , as it will be greater than all the elements currently present in the set.
Total no. of elements in $A+B$ = |B| + |A|-1 $ [$\because$ we are adding from $a_2$ to $a_n$ , i.e $|A|-1$ elements]
$\therefore$ $|A+B|$ $\geq$ $|A|+|B|-1$
If something is not clear, please comment.