How did "to lie" (i.e lie about something) and "to lie" (i.e. lie down) end up being spelled the same way?
Solution 1:
Taken from Online Etymology Dictionary
lie (v.1)
"speak falsely, tell an untruth," late 12c., from O.E. legan, ligan, earlier leogan "deceive, belie, betray" (class II strong verb; past tense leag, pp. logen), from P.Gmc. *leugan (cf. O.N. ljuga, Dan. lyve, O.Fris. liaga, O.S., O.H.G. liogan, Ger. lügen, Goth. liugan), from PIE root *leugh- "to tell a lie."
lie (v.2)
"rest horizontally," early 12c., from O.E. licgan (class V strong verb; past tense læg, pp. legen) "be situated, reamin; be at rest, lie down," from P.Gmc. *legjanan (cf. O.N. liggja, O.Fris. lidzia, M.Du. ligghen, Du. liggen, O.H.G. ligen, Ger. liegen, Goth. ligan), from PIE *legh- "to lie, lay" (cf. Hittite laggari "falls, lies," Gk. lekhesthai "to lie down," L. lectus "bed," O.C.S. lego "to lie down," Lith. at-lagai "fallow land," O.Ir. laigim "I lie down," Ir. luighe "couch, grave"). To lie with "have sexual intercourse" is from c.1300, and cf. O.E. licgan mid "cohabit with." To take (something) lying down "passively, submissively" is from 1854.
You can see that both words have different roots (legan v. licgan) but converged together in terms of spelling. It is apparently, totally coincidental as you can see there is no similarity in meaning or prior spelling along the way.
Solution 2:
To some extent, the question arises from looking at the situation backwards. An example: somebody experiences a coincidence or event that should happen as a rarity, and their reaction is "what are the odds of this happening?"
The odds of it happening to a specific person at a specific time might be infinitesimal, but the odds may be virtually 100% of it happening sooner or later to somebody, somewhere, sometime. In reality, it would be improbable for it to not ever happen. That person just turned out to be the somebody to whom it happened.
English has tens of thousands of words. They are formed from a limited number of letter combinations and evolution is generally in the direction of dropping letters to shorten words. The laws of probability predict that some words will end up spelled the same.
English users don't have insurmountable problems when different words are spelled the same, so there isn't a pressing need to force words into different spellings to keep them unique, or to prohibit the dropping of letters that allow different words to become spelled the same.
There are many words that have become spelled the same. You can select a particular pair and ask why, as if it is surprising that those two words ended up spelled alike. But the reality is that there isn't much special about any example you pick. Natural forces of language produce such examples, and random actions of history play a big role in which words end up that way.
Solution 3:
Have a look at German where the verbs are "liegen" and "lügen" (to tell something untrue"). They are very similar. So I would not be astonished that in English they have the same form. If such things happen then the sentence structures of the two identical words is so different that there is no danger of misunderstanding.
We have the same case with German "hatte" (had, normal past) and "hätte" (had*, past subjunctive). In English the two verb forms are identical and this can lead to real misunderstanding. To avoid misunderstanding English uses had* only in a very restricted way (e.g. after "if") and has replaced had* by "would have".