Here we find another reason why people find major dietary changes like this unrelatable. We have a headline that says people wrongly think that plant based diets are more expensive, and a comment section largely consisting of comments indicating that we should in fact assume a complete change of cooking habits is something we don't need to factor in while making this statement. It's a massive oversimplification.
Many of the cheapest and most filling foods available at any given grocery store or especially at any given convenience store or restaurant contain some sort of animal product, whether that be meat, dairy, or egg. Yes, if you have access and the inclination you can get a lot of ingredients pretty cheaply, and yes raw ingredients tend to follow the pattern of plant based ingredients often being cheaper. But to say that a plant based diet is less expensive is as devoid of nuance as saying it's more expensive.
If you're eating a plant based diet exclusively, you certainly do lose access to a wide range of cheap and easily prepared foods. The cheap version of a plant based diet requires either a fair bit of preparation or eating a lot of rice and beans. The version that likely most closely mimics the previous diet of people who aren't doing a ton of cooking is likely to be more expensive.
This also assumes access to fresh produce and both the facilities and the energy and ability to cook on a regular basis. The cost difference for someone who can't or simply doesn't cook every day to have a plant based diet with substantial variety is in fact significantly more expensive.
All that said, it is much more attainable to shift to a diet that leans more heavily on plants while spending about the same amount and not denying oneself access to cheaper foods that might be available or quite as drastically changing one's habits.