Well, first let's look at the history and usage of dental fricatives. This was a very common sound in Old English -- indeed, in all the Germanic dialects at one time. It disappeared from the continental varieties of West Germanic, which is why "Neanderthal" is pronounced, and sometimes spelled, "Neandertal." The "th" in Thal ("valley") is pronounced as "t" in German. This is also why Thomas is spelled as it is. Although German lost the dental fricative, it remained in the North Germanic languages and in English.
The dental fricative had its own letter in both the Elder and Younger Futhorc (runic alphabet), which was used to scratch or engrave inscriptions on stones, swords, and personal items. That letter, as it has come down to us, is called "thorn" and its runic letter is þ. When capitalization was invented, the capital form became Þ.
Thorn was originally the voiceless dental fricative (as in "thin"). But, of course, English also has a voiced dental fricative (as in "then"). They developed a single letter for that one, too, called "eth" (ð). Eth was shaped as a crossed "D," which becomes even clearer when capitalized (Ð). This same sound appears in Welsh, where it is represented by a double D, as in "Myrddin."
But what do we do when our weird English rules cause us to change voiceless into voiced fricatives? Consider the adjective loath (sometimes spelled loth). "I was loath to consider the possibility" is pronounced with the "th" as in "thin." But when we change the adjective to a verb, we say, "I loathe him," where the "th" is pronounced as in "then." (We do the same thing with "breath" and "breathe." It's the vowel change from short to long in the verbs -- represented by the silent E -- that provokes the switch.) We don't change letters to represent the sound change, nor did they in Old English. Which meant that over time, the letters þ and ð became almost interchangeable. But at least it only took one letter to represent the sound, whichever it was.
So why does it take us two letters to spell the sound? Because the old runic alphabets, made for carving, died out. Literacy arrived with the missionaries, and people began to write longer works in pen on parchment and vellum. The missionaries taught the use of the Latin alphabet to represent English. And guess what? Latin doesn't have dental fricatives.
In attempting to represent Germanic sounds with a Latin alphabet, the scribes made some adjustments. They invented the letters G, J, and W. They eventually imported Y from Greek Upsilon (Υ). So why didn't they create a new letter for dental fricatives?
Ah, but they didn't have to! They continued to use Thorn until well into Middle English. Thorn is the letter mistaken for Y in signs reading "Ye Olde" this and that. That word wasn't "ye," it was "the" spelled with a Thorn (Þe) and pronounced as such. They even used it in abbreviations, as when they wrote (ꝥ), which represented "that" -- one of the commonest English words -- thus saving much labor.
But eventually, all the extra letters -- including ash (æ), yogh (ȝ), and wynn (ƿ) -- died out. English now used only the scribally modified Latin alphabet to spell all its sounds. And the scribes had taken to spelling anything breathy or hissy with added aitches (h). So the initial sound in "should," which was spelled in Old English as sceolde, switched from "sc" to "sh." Hwæt became "what." Greek Φ was transliterated "ph," as in "phonics." Old English ceorl became "churl," and was pronounced pretty much the same. "Gh" replaced yogh and has plagued us ever since ("though," "enough," "hiccough," "ghost," etc.). Thorn (and eth) became "th."
And there you have it: a thorny bit of orthography, explained.