This is a great question. English, unlike a lot of other languages in the world, has complicated tenses in the present, future and past, that make for subtle differences in expressing thoughts. That's the case with the question you've asked here. Each one of the examples below outlines how each way that we phrase the thought can give a slightly different feel to the sentence.
"I'm interested to know what you meant by that question, sir."
- Using the infinitive form "to know" (infinitive meaning that it is not conjugated to the subject of the sentence, or it appears in the form you'd find it in the dictionary) has a nuanced implication that "I" have no knowledge of the explanation or motive that will follow with the answer. I want an answer, and I want that answer to be complete and end with my knowledge.
"I'm interested in knowing your opinion on this."
- This example uses the present progressive conjugation to follow the "I am" clause. If I am interested in knowing, the implication, however subtle and nuanced, is that I'd like to begin to understand and continue moving along with that understanding, discovering more, in an ongoing sense.
"I would be interested to know something about your work experience before we go further."
- Our final example is a bit different. We cannot use the present progressive after using the future conditional tense. One is future, one is present. Those two tenses cannot exist in the same sentence without some sort of transition in the timeline. Therefore, we have to use the infinitive version of "to know" after using a conjugated future tense.
These all convey basically the same thoughts. In each, the subject, "I", is expressing interest in some sort of knowledge about the object of the rest of the sentence. In some contexts, like the first two examples given, the phrasing is basically the same, with little difference between the two. "Knowing" something, and "to know" something, have slightly different meanings.
The changes are small and barely noticeable, but they carry an implication. For my part, I'd phrase each as shown.
