I'm assuming this is an electric stove. Most electric elements just plug in. Look for where the end of the burner coil goes into the stove, and pull up and out opposite that point, and it should come out with a gentle pull. You'll then be able to take the burner pan out as well.

I've just looked at our current stove, and it has a separate pan and ring around the burner, and those come out without removing the element, just tilt it up and you can get those out.

You also can most likely tilt up the stovetop itself to be able to clean out underneath it, so try pulling up on the front and see if it'll come up.