Why is convective precipitation associated with heavy rains?
Before we go any further it would be helpful to review this Q/A-Are clouds a gas, liquid, and/or solid?

The key to understanding this question is to realize that it is missing a comparison. Does convective precipitation produce heavier rains in comparison to some other form of precipitation ?

When we look at the different types of precipitation and the respective scales involved hopefully we can see why convective rains produce heavier rainfall. 

So there are two forms of precipitation - convective and stratiform. 

I am going to provide a definition of both types of precipitation based on the magnitude of the mean vertical velocity of an atmospheric volume. So the mean vertical velocity(obtained from observations) is the average vertical velocity over the atmospheric volume being studied(in this case the cloud(s) in question). 

So stratiform precipitation is defined as 

$ w &lt;&lt; V_{ice,typical} $ and $w &gt; 0$.  where w is the mean vertical velocity and $V_{ice,typical}$ is the terminal fall velocity of ice crystals and snow(typically in the range  1-3 m $s^{-1}$)

Convection precipitation is defined as 

$ w \approx V_{ice,typical} $ and w is in the range  1-10 m $s^{-1}$ and sometimes exceeds the value of $V_{ice,typical}$

These definitions are due to Henry Houghton as shown from the reference - On Precipitation Mechanisms And Their Artificial Modification. Also stratiform precipitation occurs mostly from nimbostratus clouds and convective precipitation occurs from cumulonimbus clouds.  

Now let us list the different types of instability in the atmosphere and the scales at which they evolve(any cloud has to form due to an instability in the atmosphere) 

1) Conditional instability/Convective instability - (usually less than 10 km horizontally) and time scale - few minutes

2) Symmetric instability - ( 200 kms horizontally) and time scale - few hours

3) Baroclinic instability ( less than 2000 kms horizontally) and time scale - maximum 2-3 days. 

Now let us match the speed of the vertical motions in the atmosphere with the scales at which a particular instability evolves. I am going to exclude symmetric instability  from this discussion. 

Frontal systems(baroclinic instability), systems affected by topographic lifting and large scale horizontal convergence produce stratiform precipitation and continuous rain. In this case growth  occurs by deposition of vapor particles and aggregation of ice particles. Because of the slow updrafts we typically see continuous rainfall spread over a large distance and slowly varying over time. 

Localized rain associated with either conditional instability or convective origin produce updrafts of several meters per second. There is not enough time for full growth of precipitation particles. Initial growth occurs just above the cloud base 
and since the time for growth is very short and rain drops grow mainly by accretion of rain drops. Heavy rainfall realized over a short period is balanced by the smaller horizontal scale of the event. When convective rainfall is observed over a large horizontal area usually it is balanced by areas that have a) either no rain b) areas having stratiform precipitation. 

References

Cloud Dynamics by Robert Houze, Chapter 6