Webster the 10th lists the etymology of this word simply as 'International Scientific Vocabulary' without explaining the roots. That's okay, though, because they should be obvious to any English speaker: 'meso-' comes from the Greek noun 'mesos', meaning middle. 'Derm', as you well know, means 'skin'. So 'mesodermic' is an adjective ('-ic') meaning 'of or relating to the middle of the skin'. That was easy. Only, it's not actually part of the skin. Sorry.
The mesoderm this adjectivises is actually the middle of the three germ layers in embryos. The outer layer will turn into skin, but the middle layer develops into muscle, bone, cartilage, and similarly useful stuff. It's also called the mesoblast, so one can conclude that 'mesodermic' means the same thing as 'mesoblastic'
Now let's jump back to that 'derm' thing. I said it means 'skin', but didn't say from whence it came. Webster the 10th (as well as American Heritage the 4th) lists it as from New Latin from Late Latin '-dermis'. Maybe so, but I think we can go back a bit further, all the way to Ancient Greek's 'derma, dermatos', still meaning skin. There's no Latin involved. Why would there be? 'Meso' is Greek, and so should be 'derm', by golly! The venerable Webster 1913 agrees with this etymology.
Etymology aside, here's the definition:
Pertaining to, or derived from, the mesoderm; as, mesodermal tissues.
Synonyms: mesoblastic, mesodermal.
The Greek & Latin Roots of English (3rd ed.) by Tamara M. Green