Argthtjtr

I was playing MTG with my friend. I attacked with a 6/4 and he blocked with a 5/5. He says my 6/4 should die because its toughness is less than its block even though my creatures attack (a 6) is bigger than his creatures toughness (5).

Unless one of those creatures has First Strike or Double Strike, they will deal combat damage simultaneously. The following will happen:

• Your 6/4 will deal 6 damage to your opponent's 5/5. (Note that 6 ≥ 5.)


• 
  At the same time, your opponent's 5/5 will deal 5 damage to your 6/4. (Note that 5 ≥ 4.)


• State based actions kick in, see that both creatures have damage greater than or equal to their toughness, and both creatures die.

This stuff is explained very well in every core set starter box by about turn 3, usually having a 2/1 block a 1/1 or similar, but let's be frank and dig out the rules. Open at 510, we skip parts that are not relevant:

510. Combat Damage Step

510.1a Each attacking creature and each blocking creature assigns combat damage equal to its power. Creatures that would assign 0 or less damage this way don’t assign combat damage at all.

510.1c A blocked creature assigns its combat damage to the creatures blocking it.[...]

510.1d A blocking creature assigns combat damage to the creatures it’s blocking.[...]

510.2. Second, all combat damage that’s been assigned is dealt simultaneously. This turn-based action
doesn’t use the stack. No player has the chance to cast spells or activate abilities between the time
combat damage is assigned and the time it’s dealt.

[...]

510.4. If at least one attacking or blocking creature has first strike (see rule 702.7) or double strike (see
rule 702.4) as the combat damage step begins, the only creatures that assign combat damage in that
step are those with first strike or double strike. After that step, instead of proceeding to the end of
combat step, the phase gets a second combat damage step. The only creatures that assign combat
damage in that step are the remaining attackers and blockers that had neither first strike nor double
strike as the first combat damage step began, as well as the remaining attackers and blockers that
currently have double strike. After that step, the phase proceeds to the end of combat step

Ok, what does that mean for us?

• Step 510.1A & C say, that the 6/4 deals 6 damage to the 5/5 in absence of first strike, double strike or protection from the color/type of the 6/4 or an effect that prevents combat damage.


• Step 510.1A & D say, that the 5/5 deals 5 damage to the 6/4 in absence of first strike, double strike or protection from the color/type of the 5/5 or an effect that prevents combat damage.


• Step 510.2 says, we got a 6/4 with 5 damage and a 5/5 with 6 damage. In both cases, the damage is equal to or higher than their toughness.
  
  
  
  
  • Both creatures get destroyed^{1 2}.
  
  
  • The destroyed creatures leave the battlefield³ and enter the graveyard⁴
    
  
  
  
  

^{1 - in absence of 702.12. Indestructible A permanent with indestructible can’t be destroyed. Such permanents aren’t destroyed by
lethal damage, and they ignore the state-based action that checks for lethal damage
2 - 701.14. Regenerate can apply: “The next time [permanent] would be destroyed this turn,
instead remove all damage marked on it and tap it. If it’s an attacking or blocking creature,
remove it from combat.
3 - might trigger as this card leaves the battlefield effects
4 - might trigger as this card is put into the graveyard effects}

I think the source of your confusion is in what the stats of a creature are (I had the same confusion myself way back when I first started playing Magic).

A creature's stats are power/toughness, not attack/defense. So a 6/4 has 6 power and 4 toughness.

Power is how much damage it deals. Toughness is how much damage it takes to kill it. When asking which creatures die in combat, it does not matter which is the attacker and which is the defender. Each creature simultaneously deals damage to the other (based on its power), and then each creature that has taken damage greater than or equal to its toughness dies.

In the situation you give in the question, each creature has power greater than or equal to the toughness of the other, so they will kill each other (baring abilities like first strike, double strike, protection, regeneration, indestructible, etc).