A field itself is a kind of concept, yes. However, they're certainly real -- it's just a way of thinking about things.
Imagine a real field, in the countryside somewhere.
It's empty at the moment, just soil. You want to sew some seed, and so get a few seeds and drop them in one place somewhere. You then do this every foot or so, so that if you were looking at the field from the air, you'd see a kind of grid of clumps of seed.
The 'strength' of this field at a certain point is simply the number of seeds at that point. You can think of EM fields the same way -- the strength of it can be different in different places.
Make more sense?