1. There will be one or two rubber seals / O rings inside the part where it swivels (Upper centre of tap), and possibly a nylon ring or split ring.
2. They leak from there when the seals have worn.
3. You could replace the seals if you know the make and model of tap. Obtain seals. Turn the taps off and remove the spout. May have a grub screw or Allen grub screw usually located round the back. Your second picture shows what may be a hole for the grub screw. Replace seals. When reassembling, if it has a split white nylon washer, the split normally sits where the grub screw comes through.
4. If you don't know what taps they are, you could try removing the spout, as above. Then measure the existing seals and try and find replacements. Not easy, particularly if they are not O rings.
5. As a temporary measure, remove spout, remove seals. Wind a few turns of PTFE tape into the grooves in the base of the spout, trying to keep them as flat as possible. Replace seals and reassemble. The PTFE can make up for a small degree of wear. This is NOT a lasting solution.
6. Replace the taps altogether. If you are not doing the work yourself this will probably be the cheapest option. Spending £80 / hour for someone to mess around with seals will soon cost more than the taps and fitting.