I would think they would jump out and flop around until they hit a source of water. If you have animals like water buffalo or whatever they have over there milling around waterways you could feasibly end up with churned up ground where puddles form.
Of course this isn't the preferred habitat of bettas, but I could see how it would happen. However, most fish will instinctively seek out larger bodies of water and usually only end up in puddles because of bad luck or dry weather.
Here is a video of killifish eggs hatching in an elephant's footprint
Even those fish though ended up leaving the puddle once they were big enough.