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The electrodes electrically connect to the skin in order to pick up tiny signals produced by the heart muscles.

Electrically you're a bag of saltwater.  Your skin (the bag) has a relatively high resistance.  The innards (the salt water) has relatively low resistivity.

To get good signal to noise ratio of something making small signals inside the bag, you want to minimize the resistance thru the bag itself.  That gives you better coupling to the source of the signals, and lowers the impedance, which attenuates noise pickup.  Piercing the skin and connecting directly to the innards would work, but would be uncomfortable and expose patients to additional dangers.

The best compromise is to connect electrodes in such a way as to minimize the resistance thru the skin.  That's done two ways: By using a large surface area, and by adding goo that makes good electrical contact and actually lowers the skin resistance.