What phenomenon causes the increase of drag with rain?
While Peter Kämpf already commented the effects I'll take a look at the numbers. Don't expect this calculations to be exact predictions, rather worst cases to know the magnitude of following effects:


Mass increase due to rain drops
Forces due to drops hitting the wing
Early tripping of the boundary layer


After deducing some equations I'll feed them with the data of two different airplanes.

Mass increase due to rain drops

The dew layer you find on your airplane in the morning can reach a maximum altitude of 0.8 mm. Raindrops, which do not fill the complete wing surface, don't usually exceed 2 mm diameter.Therefore taking an average distributed height of $h_W=1mm$ for the water on the wing should be still a worst-case prediction.  

$$m_{Drops}=\rho_{W} \cdot A\cdot h_W$$

$A$ is the projected area of the aircraft on the horizontal plane and $\rho_W$ water density.

Forces due to drops hitting the wing

Normal rainfall intensities are about 5 mm/hour and we already speak about violent rain with precipitation rates beyond 50 mm/hour. But let us take the heaviest rate ever measured: $I_R=38mm/min$.

Drops fall normally with $v_{Drop}=10m/s$.

To simplify the calculation I'll assume vertical rain and that once a drop hits the airplane it sticks to it. In reality you don't accelerate every drop to the airplane speed nor slow its vertical speed down to 0.

i) Drag increase

$$ \Delta D= \rho_{W} \cdot I_R \cdot A \cdot v_{airplane}$$

ii) Lift decrease

$$ \Delta L= \rho_{W} \cdot I_R \cdot A \cdot v_{Drop}$$

To calculate the total drag ($D=C_D\cdot \frac{\rho_{Air}}{2} V^2 \cdot A$) I've used $C_D=0.035$. $V$ is the cruise speed of the aircraft.

Early tripping of the boundary layer

While it's possible to find airfoil polars in rain searching the Internet, I'll first look at a "bugs influence" diagram where you can see the effect of small disturbances at the nose without other effects overlayed. Bugs and raindrops are similar in size, therefore the comparison should be valid.



Source

The polar above shows how the drag can double (100% increase)just because of some small disturbances at the leading edge. Keep in mind that in modern profiles the effect will probably be smaller. The next polar is from a transport aircraft airfoil.



Source

While not so important as in the glider airfoil, the drag coefficient increase (experimental results) is everywhere at least a two-digit percentage.  

Airplanes

I've taken two airplanes with really different wing loadings: A1 would be something between the two Solar Impulse models and A2 a single engine four-seater like a Cessna 172. The last three rows give the drag, lift and mass difference as described above as a percentage of its total value.

|                       |            A1 |     A2 |
|-----------------------+---------------+--------|
| Mass m [kg]           |          1600 |   1000 |
| Area A [m^2]          |           210 |     20 |
| Wing loading [kg/m^2] |           7.6 |     50 |
| Cruise speed V [m/s]  |            20 |     58 |
| Lift L [N]            |         15696 |   9810 |
| Drag D [N]            |          1800 |   1323 |
| Drops mass mW [kg]    |           210 |     20 |
| Drag increase dD [N]  |          2660 |    703 |
| Lift decrease dL [N]  |          1330 |    127 |
|-----------------------+---------------+--------|
| dD/D [%]              |           147 |     53 |
| dL/L [%]              |           8.5 |    1.3 |
| mW/m [%]              |            13 |      2 |
|-----------------------+---------------+--------|


SUMMARY

For a normal GA airplane the water mass on the wings and the lift decrease influence due to raindrops hitting the wing even considering the worst possible conditions are low single digit percentages.

While drag increase due to drops hitting the wing is relevant in the extreme case I took here, under normal conditions it should be below single digit percentages too.

On the other hand the drag increase due to the early tripping of the boundary layer is at least a two digit percentage, at high lift coefficients even more.

Airplanes with lower wing loading are more affected with rain precipitation.