Plants naturally pick up heavy metals from the soil they grow in, generally these are rather small amounts and both humans and animals can process them. There is almost no danger in consuming plants unless the soil is dangerously contaminated (generally an industrial source, or occasionally a fluke a geography).

The problem comes with the concentrated protein supplements, as it also concentrates the contaminants. Protein supplements are generally sourced from the fruit of the plant, e.g. the bean from soy or the pea from pea. This is also where much of the soil nutrients bioaccumulate, as the plant is sending a bunch of water to the fruit in order to make it grow. When millions of soybeans are then ground up and concentrated into protein powder, the lead/cadmium/arsenic/Mercury remain behind in the powder - still in low amounts, but enough that if someone is using large amounts of the supplement daily they can be ingesting a lot more heavy metals than they are aware of.

With animal-sourced proteins, contamination is generally lower (although plenty of brands still have concerning levels) simply because the protein is sourced from places where heavy metals don't preferentially accumulate. E.g. lead bioaccumulates in bones and teeth, cow-sources protein is generally whey (from milk) or more rarely from the muscles - both places naturally lower in lead Owing to the cow's biochemistry.

For the record, I am a vegetarian (vegan + eggs) and use vegan protein supplements. I buy from brands which publish third party testing results on heavy metals contamination by lot to help control this exposure risk.