Generally speaking the claim is that significant amounts of agricultural energy go into growing grain for animals, and that domestic animal populations result in methane emissions, so eliminating the animal populations means less methane and you'd also need less agricultural production as everyone would eat vegan diets, if I've spelled it out adequately.
However, this doesn't change the need for electricity generation and transportation fuel to move food and materials around the world, or from farms to cities, or for industrial production and heating and cooling, so that's where the electric vehicles, solar panels, wind turbines and storage systems are needed so we can phase out the fossil fuels from the energy mix. Yes, that's going to be cheaper and more sustainable than trying to expand uranium fission.
Furthermore, in the pre-industrial era in North America for millenia there were large populations of bison, estimated variously at perhaps 50% of the domestic cattle population (which is where the biogenic methane is coming from). For example:
(2012) "Historic, pre-European settlement, and present-day contribution of wild ruminants to enteric methane emissions in the United States"
Overall, enteric CH4 emissions from bison, elk, and deer in the presettlement period were about 86% (assuming bison population size of 50 million) of the current CH4 emissions from farmed ruminants in the United States.
There are plenty of decent arguments for eating less meat, but I don't think 'fixing the climate' is really a very strong one.