Public declaration
Without a recipient, tell can mean making a public declaration. Perhaps you could understand an implied recipient: "the world". The wider context of that sentence:
I am forced into speech because men of science have refused to follow my advice without knowing why. It is altogether against my will that I tell my reasons for opposing this contemplated invasion of the antarctic—with its vast fossil-hunt and its wholesale boring and melting of the ancient ice-cap—and I am the more reluctant because my warning may be in vain. Doubt of the real facts, as I must reveal them, is inevitable; yet if I suppressed what will seem extravagant and incredible there would be nothing left. The hitherto withheld photographs, both ordinary and aërial, will count in my favour; for they are damnably vivid and graphic. Still, they will be doubted because of the great lengths to which clever fakery can be carried. The ink drawings, of course, will be jeered at as obvious impostures; notwithstanding a strangeness of technique which art experts ought to remark and puzzle over.
suggests throughout that the speaker is explaining his decision to disclose certain information to the general public, which he'd prefer to keep limited to a small group of people within "men of science". All the phrases I marked in bold refer implicitly to the public world:
speech before the public
tell my reasons to the public, to the world at large
warning the public
Doubt of the real facts: inevitably, many members of the public will doubt the real facts; public doubt
reveal them publicly
if I suppressed this information, i.e. kept it from becoming publicly known
seem extravagant and incredible in the eyes of the public
will count in my favour in public opinion
jeered at by many members of the public
Comparable usage
As is typical in English, the verb tell doesn't work by exact rules so much as by "echoing" prior use in familiar situations. Cues like context and what sorts of objects it has or lacks guide listeners to reasonably adjust what it meant in a similar context to what you want them to see or imagine in the present situation. Here are some familiar uses of tell without a recipient that shed light on how a fluent speaker understands it in the sentence by H.P. Lovecraft.
"To tell a story" suggests reciting the story before a group of people, perhaps sitting around a campfire—or at a TED talk. Of course you could also tell a story to just one person, like a child at bedtime. As I said, this works by reasonably adjusting what the word meant in a similar context to suit another; not all familiar contexts are relevant. Sometimes people want to "tell my story", by which they mean publicly narrating a struggle they went through, so the whole community will understand or appreciate them or so that people will learn some valuable lesson.
Little kids say "I'll tell on you" to mean that they will inform someone in authority, such as a teacher or parent, about some wrong thing that "you" did. This tell without an object also suggests making private information public, or at least releasing it to someone not within the circle of those who know the secret. Adults say (playfully) "I'll tell" or "You won't tell, will you?" A tell-all (a noun) is a book that reveals private information, usually about a celebrity, usually involving embarrassing activities or love affairs.
Here's a more distant usage: "You can tell that Nicole is from Canada by the way she says 'sorry'." The speaker means that you can discern Nicole's origin from her accent, and expresses this indirectly by saying that you could state what you discerned. People understand "can tell" as a synonym for "discern", but this illustrates how tell has an old "center of gravity" involving divulging or inferring information, to which it tends to fall when no recipient is stated. Along the same line is the use of tell as a noun for something observable that "gives away" information that is hidden or concealed: "He's lying. The tell is the way his eyes dart around as he repeats himself and embroiders the lie." As a verb in this sense, "His fidgeting tells." Similarly, about matters that we can't yet discern, "Time will tell."
