What causes intra-plate faults, such as the New Madrid fault?
Good question!  I'll lay out the prevalent idea, originally (I think) put forward by Sykes, 1978 and supported by a large body of subsequent work by many authors. There are other models, however.

In the most general sense: Intra-plate faulting is due to old, pre-existing weaknesses that are reactivated due to the plate interiors being very close to critically stressed.

In other words, they're weak points in the interior of the plate.

However, we don't fully understand why some old fault zones get reactivated while others in the same orientation don't.  (It probably has a lot to do with pore pressure and fluid movement at depth.) There's a lot we don't completely understand about intra-plate earthquakes.

Also, please don't use the term "fault line". It offends my geologic sensibilities! :)  (Faults are 2D surfaces, not lines, thus the nitpicky objection.)

Most of the crust is close to being critically stressed (breaking)

There's a lot of evidence that the continental crust everywhere is close to being critically stressed -- in other words, failing in an earthquake.  (e.g. see the various Townend and Zoback papers.) The crust isn't just critically stressed near the plate boundaries, but also away from them. This may also be true for oceanic crust, but we don't have as much data there.  

Another way of saying this is that the plates are being pushed along as hard as they can without crumpling.  

Basically, this is why things like building a dam or deep fluid injection trigger small earthquakes.  The crust is already close to failure, so adding load or increasing pore pressure often triggers small fractures.  (However, these sort of things can't trigger a major earthquake unless they're on an already active fault.  You need continuous slip along a very large surface for a large earthquake.)

The plates being critically stressed is actually is what we'd expect. We see deformation at plate boundaries whenever they collide. Therefore, we know that the stresses driving the plates are greater than the "breaking" strength of the crust.  Because stresses from a collision at the boundary will be transmitted to the interior, it intuitively makes sense that the interiors should be close to, but just below (otherwise they'd be deforming), the failure threshold. (For the complete argument, see Zoback and Townend, 2001)

Continental crust is full of old zones of weakness

The continents are very old. Much older than the oceanic crust. The continents are like rafts that keep bumping into each other. Therefore, they have lots of "scars" (i.e. old fault zones).

There's a saying in structural geology: "once a weakness, always a weakness". Fault zones are typically much weaker than the crust around them.  Once a fault breaks through, it tends to be a permanent weakness that will be reactivated even if it's not in a perfectly favorable orientation.  

Critically stressed crust + weak zones --> intra-plate earthquakes.

But why don't we see activity on all old fault zones?

Short answer, we don't really know.  It probably has a lot to do with pore pressure and fluid movement.  

Some old fault zones have intra-plate earthquakes, while others don't, even if they're in the same orientation.  Pore pressure plays a huge role in faulting (it make rocks much easier to break).  There's a lot of speculation that active intra-plate fault zones may have more fluids at depth, though explaining why is difficult.