It doesn't even have to be verbal, but through showing; many animals teach each other using that, and that goes back much further than 300K years ago. But there must've been a change in early humans that made it more effective, make it go beyond basic skills.
The other thing to consider is that they reached a point where they could gather food / survive more easily. If less time needs to be spent getting food - because they've reached a level of intelligence where they can, for example, store food for longer, or prepare / cook it for more efficient calorie gathering, or grow food, or share food / acquire more than an individual needs, etc - then there's more time left for cultural exchange, experimentation and play. That is, spend more time experimenting with a stone tool to make it better beyond the base necessity.
disclaimer: I have no idea what I'm talking about. I don't actually believe early humans spent that much time surviving, looking at apes they seem to spend a lot of time just sitting around.
The other thing to consider is that they reached a point where they could gather food / survive more easily. If less time needs to be spent getting food - because they've reached a level of intelligence where they can, for example, store food for longer, or prepare / cook it for more efficient calorie gathering, or grow food, or share food / acquire more than an individual needs, etc - then there's more time left for cultural exchange, experimentation and play. That is, spend more time experimenting with a stone tool to make it better beyond the base necessity.
disclaimer: I have no idea what I'm talking about. I don't actually believe early humans spent that much time surviving, looking at apes they seem to spend a lot of time just sitting around.