I find LINQ useful when I have a collection of some object and I'm interested in items that meet a certain criteria. Simple example would be searching for all shapes in a collection of shapes that are circles.
var circles =
from s in allShapes
where s.Type == ShapeTypes.Circle
select s;
I admit I could have just written a loop with some code in it, but I find this code shorter and easier to write, and easier to read and understand.
LINQ can also be used with databases, but I actually find I don't do that very much. To each his/her own, I guess.