Why orange carrots turn black while peeled?
I must admit that my answer contains a bit of speculation.
Let me start with beta carotene which is the pigment responsible for the orange colour of common carrots. It is easily and quickly oxidised by exposure to air and enzymes already present that get activated upon cutting and exposure to light. As such it looses its orange colour very quickly.
Now, browning of fruits and vegetables happens for oxidative processes when they are exposed to air. It is a common an unpleasant occurrence.
Where I do speculate a bit is on why the browning is so pronounced in the case of carrots to be even described as &quot;blacking&quot;.
I have few possible reasons that make sense to me (consider I have to deal with &quot;colours&quot; at work, if this gives a bit of credibility).

Carrots are deep orange. The dark layer of oxidised surface on the underlying orange gives us a almost black colour.

Carrots contains other deeply coloured pigments. Some are darker than beta carotene (indeed carrots originally came in purple, and some are indeed black!) and more resistant to oxidation. As such oxidation of beta carotene can reveal darker pigments (anthocyanines) otherwise unnoticed.
This is somehow similar to what happens with leaves. At first they are green due to their content of chlorophyll. Once they fall, chlorophyll has been degraded and other red and yellow pigments become visible.

A mix of the above, depending on the type and amount of pigment present in that particular carrot.



Extra: if cooking with carrots they get a green tint, this is due to a too high (alkaline) pH. For instance too much baking soda in the batter. Pigments can be sensitive to pH, too.
Based on the comment of Lorel C.
Indeed the level of tissue damaged as well as porosity are know to exacerbate the browing process.
https://www.ncbi.nlm.nih.gov/m/pubmed/10971842/
This is evident when instead of sharply peeling the carrots we clean them by scratching with a blade hold perpendicular to the surface. This is likely to be the case depicted in the photograph by OP.
Many recipes indeed recommend the use of soft brushes to clean carrots.