Wow, that's poor notation$^\dagger$ they have in the question; it doesn't distinguish which draw the events apply to properly.
Before you draw any cards you have 26 red cards, 13 clubs (and 13 spades). After the first draw is red, they say you replace it (presumably shuffling again), so you still have 26 red cards, 13 clubs and 13 spades. At the second draw there are 52 cards, of which 13 are clubs; the chance of drawing a club is $\frac{13}{52}$. (Since the first-drawn card was replaced it changes nothing; you're exactly in the situation you started with)
$\dagger$ A better notation would be: Let R_i$R_i$ be the event "a red card was drawn at the $i$th card draw" and let C_i$C_i$ be the event "a club was drawn at the $i$th card draw; then the problem is "What is $P(C_2|R_1)$?" which is useful to distinguish from various other possible questions. Ambiguous notation is a recipe for confusion.
In reply to comment:
The actual sample space for two draws is the $52\times 52$-element set consisting of all possible pairs of cards from $(A♡,A♡)$ to $(K♠,K♠)$. When you draw a red card on the first draw, you restrict attention to the $26\times 52$ set that has only red cards on the first draw.
Given that, the events with a club on the second draw are the $26 \times 13$ cases that have a red on the first draw and a club on the second draw, and the conditional probability is $\frac{26\times 13}{26\times 52}$. Note that the first draw simply cancels out of numerator and denominator, because the two draws are independent; you can simply ignore it, because it contributes no information.
[Notation that ignores that there are two draws encourages you to make the error of thinking the sample space only has 52 elements.]