The Latin H's sound, descended from Phoenician heth through Greek eta, is discussed elsewhere in this atlas; here the subject is the shape. Heth was drawn as a fence or a ladder, a set of horizontal bars between two uprights, and the earliest Greek borrowings of the letter kept that consonantal H value and its ladder-like form.
Classical Athenian Greek later repurposed the same letter, eta, as a long vowel and dropped its H sound entirely, a change that happened only in eastern Greek dialects; the western Greek alphabets that Etruscan and then Rome borrowed from had already fixed H as a consonant before that eastern shift occurred, which is why Latin kept the letter's original sound while classical Greek moved away from it.
The Roman capital H, two verticals joined by a single crossbar, is essentially the ladder shape reduced to its simplest form, and it has barely changed since. The lowercase h, formed in medieval cursive and uncial hands by extending one upright into an ascender and rounding the other side into a small arch, fixed the shape still used in printed and digital type today.
Some Romance languages that inherited H from Latin later stopped pronouncing it altogether, keeping the letter in spelling purely as a historical marker, so that French and Italian words beginning with h are, in ordinary speech, silent survivals of a sound the Etruscans borrowed from Phoenician heth some twenty-seven centuries ago.