OK, I've read the Grauniad's report on the study now.
I think that it might be a little simplistic to say what they've said - because I think that a lot of the effects they have seen may come from the way the food is cooked.
Meat tends to be roasted, fried, barbecued or grilled. All of these ways of cooking meat, especially fatty meat, have the potential to produce carcinogens, called polycyclic aromatic hydrocarbons (PAHs). PAHs are produced even more when fat spits into the heat and gets "burnt", and then the burn-products fall back onto the meat. I found out about these substances back when I was doing my degree (100s of years ago, it feels like now) and it put me RIGHT off barbecue food! Plus I stopped charring my meat, as charred meat obviously has more of the burn products. It might taste nicer, but it's more carcinogenic.
Now, they've also said that if the protein is a vegetable source, then you don't get the same effect - well no. You wouldn't. You don't tend to char-grill or roast beans in the same way, and they don't have the same composition so are unlikely to produce the PAHs.
Obviously I can't know that this is the reason they have got the results they have, but I do believe it's a confounding factor.
Oh, and IGF-1 - that's linked to breast and prostate cancer specifically, iirc - it's found in quite high amounts in lots of milk, because it's produced by the breast tissue (udders) and the bigger the udder, the bigger the milk yield (Milk yields have increased enormously in certain types of dairy cattle, namely the Holsteins, over the last 70 years) and therefore the more IGF-1 (insulin-like growth factor -1) in the milk. It's almost certainly in beef as well.
I'm still not convinced and still think it's propaganda.