How do I find the original paper of each famous theorem?
As you observe, in many cases the most celebrated results are viewed as being so widely known and diffuse that no reference is given. Yes, a bit ironic.
A way to try to circumvent that is to look at as-old-as-possible textbooks/monographs, from times within few decades of the developments you'd want to trace back. Things would seem different to those people... For example, the Whittaker-and-Watson "Modern Analysis" will give (dangit-awkwardly-footnoted-buried...) references to many things that were new then, but not now...
Jesper L-umlaut-utzen's 1984 essay on "Sturm and Liouville's..." gives many original refs.
There is an AMS-published volume "History of Analysis..." which has many original refs.
The quasi-encyclopedic two volumes edited by I. Grattan-Guiness (sp?) are marvelous, with nearly-infinitely-many original references.
(And, if you have the time/energy to double-check, Wiki!!!)
You should find a modern research paper or book citing this theorem. If it gives a reference for it (if not, try another reference), then go to the bibliography and check the reference. Repeat this process with this new reference: this should converge!
Edit: for this kind of theorem (which can be cited in advanced undergraduate courses which never give any reference like this), it's more difficult but in your case https://arxiv.org/pdf/1103.6166 may help.
Better: they also have his collected works in five volumes,
http://oskicat.berkeley.edu/record=b14596125~S1
This link ought to show the same results it showed me, a few dozen items with author Caratheodory:
http://oskicat.berkeley.edu/search~S1?/acaratheodory/acaratheodory/1%2C5%2C39%2CB/exact&FF=acaratheodory+constantin+1873+1950&1%2C35%2C
This would probably not be the very earliest publication, but some recent papers on the arXiv refer to
Carath ́eodory, C, Vorlesungen über reelle Funktionen, 1st e d, Berlin: Leipzig 1918
https://books.google.com/books?id=RV83NrXzQwEC&pg=PP1&lpg=PP1&dq=Vorlesungen+bei+reelle+Funktionen&source=bl&ots=4z4AxxWZvl&sig=fNjqEOxlvrJUtW3X886LGU-IOhM&hl=en&sa=X&ved=0ahUKEwit8aLrj9vOAhVKyWMKHcsBCIUQ6AEITDAH#v=onepage&q=Vorlesungen%20bei%20reelle%20Funktionen&f=false
which would be an entire book. There were later editions and reprints.
Here is the link at the UCB library
http://oskicat.berkeley.edu/record=b14989908~S1