A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card?

Let the two cards drawn without replacement be denoted as $(C_1, C_2)$.

1. Total number of outcomes: Drawing 2 cards sequentially from 5 distinct cards without replacement gives:

   $$N = 5 \times 4 = 20$$

2. Favorable outcomes: We require $C_1 = C_2 + 1$. The possible pairs are:

   $$(2, 1), (3, 2), (4, 3), (5, 4)$$

   So there are $4$ favorable outcomes.

3. Probability:

   $$P = \frac{4}{20} = \frac{1}{5}$$

Hence, the correct option is A.