"Please believe me, I can solve it."
我可以解决 is a reminder that just because the rules for transitive verbs in English require us to have an externally expressed direct object, as here with 'solve IT,' the corresponding verb in another language may either not be obligatorily transitive, or be transitive, but the rules of the language, permit the object simply to be implied.
In retrospect, I think "可以" is used here because this sentence is supposed to involve the concept of "permission", but the permission aspect is lost in translation. A more accurate translation would be "Please believe me. I can solve it (if you let me)."
The "can" in the vague English reference sentence "I can solve it." could be translated to any of "會", "可以", or "能", I think.