The "home plate" is an unusual construct, that is rectangular in front (two corners), and becomes a triangle in back (one corner). Here, "corner" refers to the two front ones.
"Inside" is in relation to the batter. A right handed batter will stand to the left of the plate, so that his "inside" corner is the (left) corner closer to him, and the "outside" (right) corner is the one further from him. The reverse is true for a left handed batter.
If the pitched ball stayed "totally" inside, it would be between the plate and the batter, and be a "ball." Hitting the "inside corner" means part of the baseball barely "covered" the "inside corner" of the plate for a called strike. The baseball has to go over the plate between the batter's shoulders and knees, for a "called" strike but the rule is any part of the ball over any part of the plate over the appropriate height.