There is a price increase after halving because the supply of the coins are Low and The mining reward is halved!!!!!
Currently, the supply of bitcoins increases by 25 every 10 minutes. After the halving, the supply will increase by 12.5 every 10 minutes. Regardless, the supply is higher after the halving and it continues to increase.
Well I think that this is the most precise explanation of what is really happening to halving. The price also rises because of halving cause many believes that bitcoin go to the moon during this time. So many buy and the demand increases.