Question

How did the three day Battle of Gettysburg begin?

Answer 1

In short the battle happened by accident, with small Confederate forces meeting small Union forces and not disengaging.

Explanation:

However there is much more to it. In order to understand why and how Gettysburg started you must know about what happened just before.
If you already know that skip to the ----------------------

In the Summer of 1863, the Army of Northern Virginia, under Gen. Lee goes on the offensive into Maryland and Pennsylvania. He does this to

1. Threaten cities such as Washington D.C. and Baltimore.
2. Capture supplies to help feed the Confederacy.
3. Destroy Unions ability to wage war (such as destroying the training camp near Harrisburg)

As Lee moves into Union territory, the Army of the Potomac, under Gen. Meade follows him. While Lee tries to take Harrisburg, the Union army is south of Gettysburg. Lee gets word of this and pushes south. He has three choices:

1. Head to Chambersburg, where there is little room and could be intercepted.
2. Head to Manchester, via Hanover Junction, where he could reach D.C. but is more likely to be stopped at Westminster and trapped by forces heading to Hanover Junction.
3. Head to Gettysburg where he can retreat in any of the two other directions if needed.

He decides to head for Gettysburg and expressly orders that no-one is to engage before the army is concentrated.

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This is where things fall into the hands of the lower level generals and forces. While concentrating, Gen. Heth of Early's division marches towards Gettysburg. I couldn't find out why he did this so early, some say it was in search of shoes - I dont buy that since Gettysburg didnt have any shoe factory and Early had already passed through the area. The real reason may well be unknown.

However as Heth arrives, he engages Buford, who has dismounted most of his cavalry. This fairly inconsequential skirmish garners the attention of the rest of Early's divison and part of Gen. Reynold's II Corps, both of who head to Gettysburg.
Since neither side has a clear disadvantage they don't disengage and more forces are drawn in, the Confederates to try and crush the Union forces who in turn are trying to hold this area of good strategical ground.
At this stage, both sides are too large to withdraw properly and both believe that they can win, leading to Lee committing the rest of the army, ensuring that Gettysburg would be a big battle.

Sources:

Did the Battle of Gettysburg Really Begin as a Search for Shoes?
http://www.todayifoundout.com/index.php/2015/03/battle-gettysburg-really-begin-search-shoes/

Battle of Gettysburg
http://www.history.com/topics/american-civil-war/battle-of-gettysburg

Gettysburg Animated map
http://www.civilwar.org/battlefields/gettysburg/maps/gettysburg-animated-map/

Gettysburg
http://www.civilwar.org/battlefields/gettysburg.html?tab=facts

Note I am not following an procedure on correctly citing sources