The grammatical language used in this context (relative pronoun "that") is "antecedent"; grammarians say that "firm" is the antecedent of the relative (pronoun) "that". You can however paraphrase those words by saying that "that" represents the noun (phrase) "firm", as when the concept of antecedent is being defined in a lesson on grammar. As there are no determiners used with "firm" (the zero determiner is however technically a determiner), you shouldn't use any in your comments ("a firm" or "other firms" is not what is represented).
The that-clause modifies "firms", that is, it particularizes the concept so that you are not talking any more about all firms as in "Firms are commercial organisations." (That means all firms, anywhere on the planet, and any time), but instead you are talking about certain firms having the particularity expressed in the modifying clause (this particularity being that they operate across borders).
You can replace this construction by an ing-clause, and say the exact same thing.
- Foreign leniency law passage is thus likely to affect firms operating across borders, […]
This tells you, perhaps more vividly, that you are not talking about any firms or some indefinite firms but only about at least a part of those that are able to operate across borders (and it might be all of them). This does not means that only these firms will be affected; possibly, firms of other types will also be affected by the laws.
If you wanted to say instead that all firms operating across borders are likely to be affected you'd use the determiner "the" (definite article).
Foreign leniency law passage is thus likely to affect the firms that operate across borders, […]
Foreign leniency law passage is thus likely to affect the firms operating across borders, […]
Note: "Foreign leniency law passage" is not the most idiomatic construction that you could use; "the passage of foreign leniency law" is more usual. Moreover, in the present context "passage" is not the best word; something like "introduction" is preferable (Introduction of foreign leniency law).