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I would guess that higher strength equals greater velocity.  When they see an opening, the  greater velocity can capitilize on an opening quicker when weilding mele weapons

Or maybe its just a simple as high strength = steadily being able to weild heavy weapons

First of all, may I presume that you're talking 3.5, since Weapon Finesse doesn't exist in 4e?

Second of all, I presume it was set up that way for game balance reasons.  If strength doesn't get melee attack rolls, then what else is it good for besides lifting heavy loads/doing a few athletic things?  Dexterity adds to AC in both 3.5 and 4e.

Also, don't think of strength as just muscle-bound power, and don't think of AC as just dodging an attack.  Perhaps the low strength character hits but his blows just glance off of armor, or he would get exhausted after swinging around a weapon for more than a few seconds. 

I haven't played any 3.x, but I believe the reasoning for fighters being better earlier is due to the trope "Linear Warriors, Quadratic Wizards". This means that, while fighters are better at early levels, at a certain point wizards overtake and shoot ahead of them. If you have mainly played lower level games you may not have encountered this.

I agree, the strength modifier doesn't make sense for all weapons, mainly the smaller ones. Whilst an axe clearly needs strength to wield accurately, a dagger is far more of a dex weapon. In the homebrew system I play in/run, the weapons are differentiated into strength and dex based weapons for exactly this reason. However, in D&D the game has been balanced as it is so I'd be wary about changing something so big.

Also consider that the stronger warrior will have greater ease bashing through someone's physical weapon defenses. Parrying someones weapon becomes harder when they're stronger because they strike with more force.

I agree, you would not want to house-rule that because it messes with game balance in a less than desirable way and it would require you to have to adjust every attack bonus of every monster which would also mess with their balance.

Something to consider is the "reality" of in-game combat.  A round is six seconds in duration but the combatants are not just standing around waiting for their turn to strike.  During the entirety of a combat round, combatants are slashing, stabbing, dodging, parrying, etc etc etc.  The "to hit roll" is only the culmination of one's ability to find an opponent's weak spot and take advantage of it.  And one's Strength plays a major part in the "combat dance".

Think about sword fights in movies.

The one that springs to mind is the final battle between Robin Hood (Kevin Costner) and the Sheriff of Nottingham (Alan Rickman) in Robin Hood Prince of Thieves.  Costner is wielding a longsword and Rickman a Bastard/Great Sword.  During the fight you can tell there are times when each combatant is tired and tries to use finesse (aka dexterity) and it fails miserably.  It is only when they put their muscles (aka strength) into their swings that they do any real damage or instill fear in the other.

On the other hand, think about the duel between Inigo and Rugen at the end of The Princess Bride.  They are both using rapiers.  Both of them use weapon finesse to find weak spots in their opponent's defenses and make quick jabs.  In D&D mechanics terms the two stabs Rugen puts into Inigo's pectorals would be one to hit and damage roll and at the end when Inigo runs through Rugen that would be a critical hit and damage.