They don't, in the sense that they attack them. There are two main effects at work.
1) The Grey squirrels out-compete the red ones. They are slightly larger, rather stronger and a little bolder than the native reds. They also are more conifer-adapted than the reds, and the forestry commission seemed mainly in the past to plant conifers, thus making the grey squirrel "right at home".
2) They carry a virus, the squirrel parapoxvirus that is not fatal to them, but usually is to the red.
There are other considerations too, but these are the main ones.
