While it's clear that drilling for more oil & gas would increase supply
Even that much isn't crystal-clear. Raw material sources are only one piece of the puzzle. You still have to transport the crude, refine it, and then distribute the final product. Putting in a lot of new wellheads likely means you won't have transportation infrastructure in place to get the crude to the refinery and have to use makeshifts like trucks that are low volume and expensive compared to a pipeline. This is even more complex because most of the infrastructure we do have doesn't run in those directions - for many years, US petroleum companies were selling off most domestic crude on the global market because most US stuff is high-grade, easy to refine and fetches a high price. Then importing nearly-impossible-to-refine garbage from other countries at a low price to maximize profits because our refineries can refine almost anything. So the infrastructure from the oil fields goes to shipping-out ports while the refineries are mostly fed from importation terminals. There's slack capacity in the refinery system because the US has stopped importing garbage oil from Russia and Venezuala, but it isn't necessarily enough to handle a wide increase in domestic drilling. Only when we reach the refined-product stage do we encounter an area where the existing infrastructure could probably handle it.
Once you've sunk immense capital into the wellheads, then even more into giant pipelines for distributing crude, and possibly even refinery improvements to handle the volume, you're looking at a massive red mark on the ledger. The oil itself will be inherently more expensive as ell because it probably will be high quality and will be produced under the high wages and expensive safety standards inherent to developed nations. With all the costs involved, it is very unlikely that we can actually drill our way to lower energy prices.
Meanwhile wind and solar are incredibly (and increasingly) cheap, while nuclear is expensive but produces so much power that the cost is heavily offset. The way to bring energy prices down is to invest heavily in those areas to eliminate fossil fuels from the power grid. Followed by a push for more mass transit (subsidized electric busses as well as light passenger rail) and pushing even harder to phase out ICE passenger cars in favor of EVs. That's worth doing for
purely economic reasons, with the nice side benefit that pollution would drop massively.
EDIT because I forgot half of what I was trying to respond to
(In the same interview he said his policies would reduce energy prices by 50%, and also that if Harris' tax plans take effect that a middle-class tax bill would likely quadruple. The only way I can think of a tax bill quadrupling is if you are presently sitting very close to the zero tax level, basically just over the standard deduction. So while maybe true in some specific corner cases, I doubt it would be true for the "average" taxpayer.
The people currently sitting close to the zero tax level aren't going to be affected by any proposals from Harris - she's not proposing increases to the lower tax brackets. If anything, the expanded credits and rate
reductions she wants to push would reduce or eliminate those people's tax bills. Where she wants to raise taxes is at the corporate level and on the very wealthy. Who also happen to be the ones that have lots and lots of money to fund organizations demonizing anybody who might increase their tax bills.