What are wire rope thimbles for? Will the wire rope strength be compromised if I don't use them?
They will be used on a fence.
Those are to help protect the cable. By keeping a loop and not a tight wrap around the anchor point it strengthens the cable compared to not using them. If the cable will be on something that moves not just static force they prevent wear on the cable. If a static load I may not use them but on a mobile load I do use them.
Wire rope will have a minimum safe bending radius and there will be a matching thimble for each specific diameter wire rope.
If a connection is made to a hole in a tang, the sharp (or not sharp) edge of the hole will cause the wire rope to kink or otherwise bend below the recommended minimum radius.
If the wire rope is placed under tension, the outer wires composing the rope will have greater stress and are more likely to break, while the inner wires (relative to the bend) will have a more forceful contact with the hole in the tang.
One can get away without thimbles, for example, if one is passing the wire rope around a pipe or similar cylinder with a radius greater than the minimum recommended for that particular wire rope.