While 'that' has many meanings, the meaning here is defined by Merriam-Webster.com as:
1c : the kind or thing specified as follows * the purest water is that produced by distillation
Using this definition, 'that' implies there is more than one kind or thing, and the phrase following 'that' will specify which kind or thing is being referred to. Recall the original sentence:
It is also extraordinarily difficult to meet the third criterion for causal studies, that of controlling for all other potential causal influences.
In the original sentence, 'that' is a pronoun in apposition with the preceding noun phrase 'the third criterion ...' This definition of 'that' implies that there is more than one criterion. The following phrase 'of controlling for...' specifies which criterion is being referred to.
A paraphrase could be:
It is also extraordinarily difficult to meet the third criterion for causal studies, which is the criterion of controlling for all other potential causal influences.