My view is that the best way to get promoted is to already be operating at that level. This makes it a no-brainer for the company. Rather than asking "Should we give Konerak a chance in a more senior role" you want them saying "Konerak is already effectively functioning as a senior [whatever], we should make if official".
So, your answer, if asked "why we should promote you",should be along those lines. For example: "Well I've already been doing [this] and [that] for the past 6 months and I think I've shown that [the other] in that time".
What's in it for them? Exactly what you've said - they want to reward good work and show that it's possible to build a career there. Seeing internal promotion will energise and motivate other staff to earn promotions, their team will improve and they will hold on to their best people. Plus, by hiring internally they get somebody who already knows the organisation and market space and doesn't need time to ramp up on both.