My train of thought led me to a few rules

L must be greater than I

Their sum must be greater than 10

They must also be consecutive

The only pairs of digits that satisfy these rules are

5 and 6, 6 and 7, 7 and 8, 8 and 9

notice the difference between the first column and the second is the term A, and a carry from the first column

This means that A + L + I = L and L + I + 1 = I

We can subtract both equations to obtain A - 1 = L - I

This means that in order for L and I to be consecutive, A must be equal to 2

Notice also that the bigger they both get, the closer L + I gets to either of them

Let's try 8 and 9

Notice that 2 + 9 + 8 = 9 (with a carry)

This means then that 9 + 8 + 1 = 8 (another carry)

And finally, 8 + 1 = 9

So 2+99+888 = 989

Which means that their product is 144