The history of the English orthography is a long one.
The Roman charachters as we know them today were introduced between 800 and 1000 A.D. by missionaries intent on spreading the words of their god among the then largley pagan population. The English language was a laguage from the west germainic family of languages, and the Roman glyphs were made for Latin, a language that was characterisically not from the west germainic family of languages. English had a sound with no correspondent or even soundalike in Latin. This sound was the voiceless dental fricative /θ/. What these early scribes did was take a letter from the existing FUTHARK, the writing system already in use by the Anglo-Saxons, and latinise it. This resulted in the glyph 'þ', which whilist blending into it's surroundings, is quite strange looking at a first glance, with both its ascender and descender. The story of 'ð' is much simpler. it is a roman 'd', given an extra stroke to denote it's different pronunciation.
So what happened?
The Fr*nch Happened. After Raping and murdering every living, breathing thing on the southern British Isles in the Norman Invasion, they made a point of altering english words to be spelt like Norman fr*nch was, and since the only imitation of /θ/ they cound muster was a /t/ with some aspiration on the end, they decided to write this unfr*nch sound 'th'.
You probably noticed, however, that I was able to use those letters. So what's going on?
The Icelandic Happened. Because of their almost total isolation from the rest of europe, the fr*nch did not reach this island, and these letters were preserved. Thankfully, this lasted until today. In icelandic 'þ' and 'ð' grew to be pronounced differently. 'þ' has it's usual pronunciation, but 'ð' became to voiced dental fricatative /ð/.
Most advocates for resotring both of these letters their rightful place duplicate it's dehavior in Icelandic Orthography.