The Law was not brought in (παρεισηλθεν - pareisēlthen) with the express purpose of increasing trespasses; but rather, it had that effect, since the law made clear which things were transgressions and which were not (see, e.g., the discussion you started on Romans 5:13).
Transgressions perhaps made in ignorance before the Law became more severe, since, having the Law as a guide, the transgressor should have had no doubt that they were offenses.
The verse may not seem perfectly clear because the phrase "so that" in English can indicate either a cause or a consequence. In this case, it indicates a consequence, not a cause. The same ambiguity exists in the original Greek in the word ἵνα (hina). The Byzantine Greek commentator John Chrysostom (349-407) explained:
The word “that” (ἵνα) is not every where indicative of cause, but frequently also of the event of things. Thus Christ Himself uses it, when He saith, For judgement I am come into this world; that they which see not may see, and that they which see may be made blind.1 So likewise Paul in another place, when discoursing of the law, he writes, And the Law came in beside, that the trespass might abound. But neither was the law given to this end that the trespasses of the Jews might be increased: (though this did ensue:) nor did Christ come for this end that they which see might be made blind, but for the contrary; but the result was such.2
I think the NIV would have been clearer with the simple addition of a comma (as appears in the KJV):
The law was brought in, so that the trespass might increase
The Orthodox New Testament translates the verse:
Now the Law came in beside, so that the offense abounded
1. John 9:39
2. Homily XXVII on First Corinthians