First, what are the mythological creatures? These are the winds personified as gods. The Greeks named various winds by the direction they were coming from, and these names also referred to the gods from which they emanated. Mainly these were from the cardinal directions, and had attributes that corresponded to when and where those winds they emanated blew. The north wind, Boreas, was the bringer of winter, presumably in relation to cold winds blowing down from the north in cold fronts at the end of autumn, for example. A variety of other minor winds existed as well for specific prevailing or important winds.

In the map you link we can be certain that these are the wind gods (the Anemoi) being portrayed, as they are labelled. You can see the label for "Aquilo" in the north (the Roman name for Boreas), and "Zephyr" in the northwest, amongst others. Zephyr is supposed to be the west wind, but he's been squeezed to the northwest corner by the shape of the map projection vs the shape of a rectangular surface.

That's what, but you also ask why. Certainly Martin Waldseemüller wasn't a believer in literal wind gods in 1507, nor would he believe in actual edges of the world, making simultaneously, as he does, gores for globes. Rather, it seems he's decorating the margins of the map. After the busts of Ptolemy and Amerigo Vespuchi, and the various text boxes, he's left with a bunch of odd spaces and it would look a bit repetitive with just a bunch of waves. Other renaissance maps also fill in the margins with other mythological, natural, or political scenes. But the winds are fairly common, as part of an earlier tradition dating back at least to late antiquity.

Older maps with the Anemoi have them in more regular positions. The four cardinal directions are probably the original concept, though eventually mapmakers would come to place up to 12 different winds. While the names of the winds originally were meteorological, they eventually became conflated with the names of compass directions, with eight, twelve, or even 24 distinct "winds" defining directions. Where today we might say "go northwest", the renaissance mariner might say "go towards Maestro". Different cultures and users would have different names for the winds.

By placing the winds on the map, one may remind the viewer of the map which direction is which, especially important in the days before north-up became standard. And, by the way, remind them of the names of all the directions. If you've ever seen a map with the label "North" at the top, or a compass rose with "N", you've seen the same thing. By the time we get to Waldseemüller, he knows that winds are part of what one may expect on a map, but he no longer expects them to have a practical purpose, having developed latitude and longitude and other conventions to allow the viewer to orient the map. The compass rose, originally a compact representation of the winds, would eventually come to supplant the winds as an orientation guide.

These are the twelve winds of ancient and medieval tradition (or one tradition, anyway).

There were various systems to describe the winds. Apparently originally there were only 4 named winds (e.g., in the Iliad). In his work Meteorologica, Aristotle named 10 winds - the four main north, south, east, and west winds, as well as the four in between those (northwest, northeast, southwest, southeast), but also north-northwest and north-northeast winds...but no counterpart winds from the south. Since 10 winds seemed a bit weird, his system was adapted by a later geographer, Timosthenes, into the more common 12 winds. Another system, by the geographer Eratosthenes, had only 8 winds, and this was the basis for the octagonal “Tower of the Winds” in Athens, which was probably built during the Roman period. The Romans borrowed the concept of 8 winds and made further divisions into 16 and 24 winds. But in the end the 12-wind system won out - it conveniently matched up with other things associated with 12:

“The division into twelve pass generated an essentially astronomical diagram that corresponded to the twelve hours of the day and night, the twelve months of the year, and the signs of the zodiac.” (Baumgärtner, pg 117)

In the Christianized Roman Empire, it also matched up with Biblical occurrences of the number 12 (the disciples, the Tribes of Israel, etc). So the idea that there are a specific number of winds was sort of based on actual scientific observations, but it was also influenced by religious numerology.

For the medieval world, the 12 winds were popularized by the Etymologiae, the encyclopedia compiled by Isidore of Seville in the 7th century. As the Romans had done, Isidore used some of the Greek names of the winds, but also uses Latin translation for some of the others. After Isidore, everyone learned and knew the 12 winds, but there was no real “official” list, so they sometimes have different names and positions in other medieval works. In a 12th century encyclopedia by Lambert of St-Omer’s Liber Floridus, the winds were depicted for the first time with anthropomorphized human (or human-ish) faces. The first 16th-century maps that included all of the known world, including this one by Martin Waldseemüller, also typically depicted the anthropomorphized human faces.

Clockwise from the top left on this map, the names Waldseemüller used are Zephirus, Chorus, Circius, Aparctias, Aquilo, Cecias, Subsolanus, Vulturnus Eurus, Euronotus, Notus Auster, Lybonothus, and Africus. Those aren’t quite the names used by Aristotle, Timosthenes, or Isidore, but there were lots of possible ways of naming/arranging them. The idea that there were 12 of them was more important than their specific names and locations.

Sources:

Ingrid Baumgärtner, “Winds and continents: Concepts for the structuring the world and its parts”, in Maps and Travel in the Middle Ages and the Early Modern Period: Knowledge, Imagination, and Visual Culture, ed. Ingrid Baumgärtner, Nirit Ben-Aryeh Debby, Katrin Kogman-Appel (De Gruyter, 2019)

Barbara Obrist, "Wind diagrams and medieval cosmology”, in Speculum 72, no. 1 (1997)