Why does conventional electricity flow out of + in a battery, but out of - in components?
It is like that.

Current is "attracted" to the negative end of the component. 

That is why you have to attach it to the positive end of the component... so that it will have to flow through the component to reach the negative end. 

If you connected directly, it would not flow "uphill" to reach the positive end, thus doing no work (bypassing the component).

Consider this circuit:



I've created a contrived circuit and indicated the polarity of the component's terminal closest to the top of the figure. We can draw some conclusions:

R1 and R2 are resistors, a component through which current may pass equally well in both directions (polarities). Consequently, the polarity of the component is determined by how it is connected in the circuit. In the case of R1, the top most terminal is connected to the battery negative and current moves through the component upwards. The situation is reversed for R2, current moving downwards. 

D1 and D2 are diodes. Diodes are polar. They conduct preferentially well in one direction. For this reason, we say, practically, that it has a "positive" and "negative" end. This isn't strictly true since the voltage of the "negative" end could easily exceed the "positive" end  as is the case for D2 (better terms are "anode" and "cathode").

The "positive" end of the light emitting diode is the base of the triangle (for D1, the edge top-most in the figure), the cathode (the horizontal line) is the "negative" end. So for D1, the current proceeds downwards. However, for D2, you would expect the current to proceed upwards, but this isn't possible due to the battery's orientation (current won't flow backwards into the battery when it's the only source in a static circuit).

Further, current won't flow downwards through the diode (that's backwards through the diode) as the diode's behavior prevents it (diode's only allow anode -> cathode flow). As a result, no current flows through D2 and it will not emit light as a result.

A note about polarity

It has become convention in electrical engineering to define current flow as positive to negative. EE is an applied field and progressed faster than the physics it relies on (who cares if the models are correct so long as they work, right?!). We now know that electronic currents are composed of moving electrons (negative charges) and the gaps they leave behind (positive "holes"). 

In actuality, currents consisting of negative charges are moving in the direction opposite to the way we conceptualize them... but opposite charges moving in the opposite direction has the same effect, so our previous model works just fine, which is why it has remained in use.