For the last-half of the distance to first base, a runner should not step inside the foul line, for if a fielder makes a throw parallel down the line to first, it is construed that the runner is interfering with pay.
There is a second chalk line, along the first base line, called the 45-foot line for just this purpose, defining the 3-foot wide runner's lane a runner must be within.

MLB Rule 6.05(k)
In running the last half of the distance from home base to first base, while the ball is being fielded to first base, he runs outside (to the right of) the three-foot line, or inside (to the left of) the foul line, and in the umpires judgment in so doing interferes with the fielder taking the throw at first base, in which case the ball is dead; except that he may run outside (to the right of) the threefoot line or inside (to the left of) the foul line to avoid a fielder attempting to field a batted ball;
Rule 6.05(k) Comment: The lines marking the three-foot lane are a part of that lane and a batter- runner is required to have both feet within the three-foot lane or on the lines marking the lane. The batter-runner is permitted to exit the three-foot lane by means of a step, stride, reach or slide in the immediate vicinity of first base for the sole purpose of touching first base.
Detail and examples:
So, for the last half of the 90 foot distance to first base the runner must not have his left foot even one inch on the dirt inside the chalk foul-line, until that last moment he steps over to touch the bag, because the bag is inside the foul line.
This is never a problem for runners veering out to round for second, but running out first base straight down the line – even if both feet of the runner are hitting the chalk line he is supposed to run down, but his left foot is just an inch over the line into the infield – he can be called out for interference of a thrown ball. He can be called out instantly (dead ball) even if the throw does not hit the runner, but the home plate umpire (who has the call, not the first base umpire) determines the throw was interfered with. (Note: A throw must be made for interference to occur.)
I just saw the play again (5/17/2015, Rockies -vs- Dodgers, @55:00) and even though the throw from the catcher down the line was way over the head of the first baseman - a ball that was not even catchable - and the player was never hit by the ball – the umpire still called the runner out immediately before he even got to first.
It is very odd to watch, because the runner was just going down the line like everyone does, within the 1 foot or so width of the first base plate he was running towards, but his feet were just slightly to the left of the foul line, so he was infield, and thus “outside the runner's lane”.
So basically every single runner just runs down the line – every single little league, high school, college, and evern pro game. But 1 in 1000 times a fielder has to make a throw directly down the line, this becomes interference.
I always though a runner had the right to run straight down the line, but he does not. He has to run to the right of the line down to first for the last 45 feet.
Ref.
Scenario: Dropped third strike or bunt where the thrown ball hits the runner on his way to first
Runner runs to first running "inside the running lane" (between the foul line and 45-foot line) and is hit with the ball before touching first base.
Correct call: “runner is safe“
Scenario: Runner runs to first running outside the running lane and is hit with the ball before touching first base.
Correct call: “runner is out“
(So far, so good, fairly straight forward in those two situations.)
Scenario: Fielder sees that the runner is running out of the running lane and is in the way of the throw, so he holds the ball and doesn't make the throw.
Correct call: “runner is safe“
Explanation: In the rule you'll notice that interference is "with the fielder taking the throw at first base". So if there is no throw, then there is no interference.
Scenario: Runner runs to first running within the running lane and is hit with the ball before touching first base on his last step as he moves over to touch first base.
Correct call: “runner is safe“
Explanation: The runner is allowed to move over out of the running lane to touch first base and clearly if he is moving over at the end of the running lane, his intent is to touch first base and not interfere with the throw.
Scenario: First baseman sets up in foul territory to receive the throw from the catcher on a dropped third strike. Runner runs to first running outside the running lane (to the left in fair territory) and is hit with the ball before touching first base.
Correct call: “runner is safe“
Explanation: The runner is allowed to exit the running lane in order avoid a fielder, which in this case is a first baseman who is standing directly in the running lane. Since the runner is running to the left of the running lane he is not intentionally trying to interfere with the throw or the first baseman and should be called safe.
Scenario: First baseman sets up in foul territory to receive the throw from the catcher on a dropped third strike. Runner runs to first running outside the running lane (to the left in fair territory) and then moves into the running lane and is hit with the ball before touching first base.
Correct call: “runner is safe“
Explanation: If in the judgment of the umpire, the runner made an "egregious" move where it is obvious that he is deliberately trying to be hit by the ball or interfering with the first baseman's ability to catch the ball, then the runner should be called out. But the runner is protected by the running lane and that move alone to get back in the running lane should not be construed as "intent".