I will venture a translation. I start with a very literal version, which I do not usually do (it always smacks of “see, teacher, I recognised the ablative plural” etc. to me), but in this case there are some things that I am not sure about, so I want to be explicit about how I understand things. I will then also provide a more usable translation. Quae praefati, interpretemus.
Ideal system by which is expressed the pushing out of waters through water-carrying subterranean channels from the sea and into the hydrophylacia of the mountains, and the concoction of subterrestrian waters through fire-carrying channels.
There are some difficulties here:
- protrusio: I did not find this word in any dictionary, but it was certainly formed from protrudere in the same way as the modern “protrusion,” although in meaning it is probably different, as liquids do not usually “protrude” in modern English.
- concoctus: Also not in the dictionary. What is clear is that it is derived from concoquere; what is less clear is how. It looks like the participle, but does not really fit in the sentence as such; Hugh in the comments therefore suggested it should be concoctos and then modify canales. In my opinion that makes little sense either, and I therefore suggest it is the noun concoctus, -us, a substantivation of the verb. Now concoquere literally means “to boil [several things] together,” but also “to boil thoroughly,” and from that, “to digest.” (There are a few other medical and mental meanings which I think cannot really apply here.) Now, reading the explanation below the picture, it is clear that Kircher thought the water is boiled by coming onto contact with fire-carrying channels, steam is generated and rises up to the surface and mountains. So I decided to translate concoctus as “boiling” here. This is the part I am least sure about.
- hydrophylacia: This is supposed to mean “water store,” but is a technical term of Kircher's, who theorised that there are great subterranean cavities containing liquid water. I would leave this untranslated to make clear it refers to a specific theory.
- subterraneus vs subterrestris: The first one is well-known, the second one isn't. It is unclear why Kircher uses one in the first instance and the other one in the second; I decided not to make anything out of it and translate both as “subterranean.”
Thus we get:
Schematic depiction showing how water is thrust from the sea and up to the hydrophylacia of the mountains through water-carrying subterranean channels, and how subterranean water is boiled by fire-carrying channels.
I also looked for translations online and found this one into German in Zeitschrift der Deutschen Geologischen Gesellschaft (Journal of the German Geological Society) from 1992 (unfortunately only visible through Google Books' snippet preview, so I cannot even say for certain it is a translation of this particular map):
Ideales System, das den Ausstoß der unterirdischen Gewässer durch Kanäle in die Hydrophylacien der Berge infolge Erhitzung der Gewässer durch feuerführende Kanäle zeigt.
It interprets the text more or less the same as I did above, except (a) hydragogos is lost, (b) ex mari is lost, (c) concoctus is translated as Erhitzung = “heating up,” and (d) it invents a causal relationship (infolge) between the heating and the thrust, which makes sense but is not in the original Latin.