Wait for the burners to completely cool (we can't stress this enough!), then remove them from the stove top. Most should slide out with a gentle tug and lift from the connection point, but if you have questions, refer to your manual (or find your stove's manual on).
Using a cloth and a bit of mild dish soap and water, rinse any residue from the burner coils. Take care not to get any part of the the electrical connection wet, and don't submerge any part of the burners.
If there's any cooked-on food residue that won't come off with the soap-and-water mixture, use this method: Make a paste from baking soda and water, then apply it to the tough spot. Give it about 20 minutes to sit, then scrub and rinse the burner.
Make sure the burners are completely dry before you try to replace them.