Supposedly because they contain glycoalkaloids. I'm not sure who invented the Autoimmune Protocol (AIP), but it seems to have the same scientifically questionable origins as Leaky Gut Syndrome. I have yet to see any scientific evidence supporting AIP.
There's one study hypothesizing that glycoalkaloids can increase/aggravate intestinal permeability (which is a real thing), but this study only looked at mice [1]. The hypothesis has not been rigorously tested in any serious clinical study on humans.