The original reason was to differentiate the Fighting Man from the other classes as a combat specialist
The original Strength bonus to hit, solely available to Fighting Men (what we now call Fighters/Martials) was not introduced until the Greyhawk Supplement to OD&D (1 Feb 1975, TSR).
In the first combat system, the only original bonus "+ to hit" was for Dexterity for missile fire. (Unless you had a magic weapon, for example.)
Page 11, Men and Magic, TSR 1974: +1 to hit with missiles with Dexterity
above 12 (my note: 13 Dex or greater yielded +1 to hit with missiles).
Strength above 13 gave a Fighting Man an experience point bonuses of 5%; Strength above 15 gave a 10% XP bonus, but neither gave a bonus to hit nor to damage. A Strength based bonus to hit was at best indirect -- with more Strength you'd get to a higher level sooner. A second level fighter hit AC 2 on a 17, and 4th level fighter (Hero) was the next table up and hit AC 2 on a 15, a 7th level hit AC 2 with a 12, etc. (Page 19, Men and Magic, "to hit" per the table.)
- Note: the Holmes "blue book" kept this approach. (Page 6, D&D; TSR, 1977)1.
This changed in the first OD&D supplement: Greyhawk (the most common addendum / correction / update to OD&D).
Greyhawk page 4 (devs = Gary Gygax / Rob Kuntz):
Other character types may engage in hand-to-hand combat, but only
true fighting men are able to use their Strength and Dexterity to
utmost advantage."
With that point in mind, we proceed to page 7 of Greyhawk:
"Strength also aids the fighting man in his ability to both score a
hit upon an adversary, and damage it."
"Fighters of exceptional strength are now far more formidable
opponents ..."
Then on page 8 of Greyhawk, we note that only Fighting Men
" ... with a dexterity of greater than 14 can use their unusual manual
dexterity to attempt to dodge or parry opponents attacks."
Summed up: only the martial characters could, via their profession, take advantage of their Strength to land an effective / telling blow and use their Dexterity to avoid getting hit. Anyone else could be strong, but would not get the advantage of "+ to hit." Anyone else could be agile, but since they weren't professional fighters, the Dex based armor class advantage didn't accrue to them.
Simple Answer to "Why?"
To make Fighters more formidable opponents. (Fighters tended to be melee specialists / tanks from the beginning.)
Commentary: it was understood during the play tests that went on where D&D first grew that the problem of the Linear Fighter and Quadratic Mage (as we call it now) was looming.
But that approach didn't last very long.
This restriction to Fighting Men of the Strength benefit to hit, and to damage, was lifted in the 1e AD&D PHB. The "Why" on that is another matter, given the speed at which the game was growing.
- BECMI applied a less complex version of that same bonus for high scores in abilities, per page B7 of Basic D&D, Moldvay (Mentzer), TSR, 1980. Str bonus for hit and damage was +1, +2, +3 for 13-15, 16-17, 18 respectively. Similar bonuses were applied to other attributes. That score-to-bonus relationship can still be found in 5e, for all attributes, albeit with a few refinements.
While only Fighting Men and their sub-classes could benefit from Extraordinary Strength (18 Str Fighters rolled percentile dice 01-00 for additional benefits) in AD&D 1e, the stronger-than-average Cleric would do a bit more damage when he whacked someone with a mace. The AD&D 1e system formed the backbone of what we play to this day, for all of the variations that have come to pass. (As did BECMI, which took a similar approach.)
The "+ to hit" and "+ damage" based on stat ability is rooted in making the martial character stand out in combat. This makes a lot of sense, when you begin to apply real world comparisons between people who really know how to fight (professionals) versus anyone else.
1 Of passing interest, but off topic for this answer, the cover of the book says "The Original Adult Fantasy Role Playing Game for 3 or more players."
