The paper uses rather dense language so I can understand why you might be confused.
The wikipedia page offers a good example of the Wason selection task:
You are shown a set of four cards placed on a table, each of which has
a number on one side and a colored patch on the other side. The
visible faces of the cards show 3, 8, red and brown. Which card(s)
must you turn over in order to test the truth of the proposition that
if a card shows an even number on one face, then its opposite face is
Wason designed the task to examine the human tendencies to make four kinds of inferences. Two valid inferences: modus ponens (if P implies Q and P is true, then Q is true), and modus tollens (if P implies Q and Q is false, then P is false). And also two invalid inferences: denial of the antecedent (if P implies Q and P is false, then Q is false), and affirmation of the consequent (if P implies Q and Q is true, then P is true).
Both experiments in the Wason paper manipulate between an experimental and control condition the instructions and preparation given to participants before they perform the selection. This manipulation defines the independent variables of the experiments that you're asking about.
In the first experiment, subjects in the experimental group are prompted after trying the selection task to "project falsity", or, in other ways, to say what values associated with P, not-P, Q, not-Q in the context of the selection test would make a conditional sentence false. Subjects in the control group were instead merely prompted to think again about their initial selections (ie revise them) "because people often do this task too quickly and get it wrong". Since Wason investigated how these changes in instructions affected behavior, they are the experiment's independent variable.
You can see from this quote here that something similar is going on in the second experiment:
In the experimental group the conditional sentence was presented and
the four programme cards were handed to the subjects who were asked to
pick out “the one card which makes the rule false” (i.e. falsifies the
conditional sentence). They were then asked to pick out any which
“prove the rule true.” It was explained to them that their decisions
meant that the converse of the sentence could not be assumed-“that the
rule only held one way.” The subjects in the control group were given
a similar amount of time to understand the conditional sentence without any
explanation of its meaning.
The special manipulation in the instructions here is that experimental subjects 1) were told that only one card would make the conditional statement false, and 2) received some explanation that the converse of said conditional statement couldn't be assumed. So, they got extra information and Wason examined how it impacted their answers. That's the independent variable for the second experiment.