Using oversampling to increase resolution of a DC-signal as input
My input signal is a dc signal (sensor output) and im getting kind of a headache to understand why oversampling can increase the resolution of dc signals.

The ADC puts out integers.  So, let $x$ be the integer that would come out of the ADC (100.3, say, or -333.3).  Now let $y = \lfloor r \rfloor$ be the quantization operation, where you get straight integers, like 100 or -334.
Now, let $r$ be the sum of some actual value $x$ and some nice well-behaved zero-mean noise -- $r = x + n$.  For the right sort of noise (zero-mean Gaussian that's big enough is one kind) the ADC output is a random variable with a mean equal to $x$.  So treat $y = x + n_{ADC}$, where $n_{ADC}$ is a zero-mean noise process.
Now when you average a bunch of samples of $y$, the effect of $n_{ADC}$ on the signal diminishes (in fact, its deviation goes down as $\sqrt n$ for an $n$-sample average).  That's why oversampling and averaging reduces the noise and increases resolution.

How is the low pass filtering achieved?

In the method you outline, you're averaging the input samples.  That's effectively low-pass filtering.
Note that the app note gets it slightly wrong: it's having you add up $2^n$ samples, then shift by $n$, effectively dividing by $2^n$.  However, by virtue of the averaging, you're effectively adding bits to the ADC resolution*.  Usually when I do this I just take the sum of the $2^n$ samples, knowing that there's a lot of noise in the LSBs.  If I do shift down, I shift down by $n/2$, not by $n$.

Is something like SNR a good specification to describe the quality of a dc signal?

Not really.  SNR is good when you have a well-defined signal strength.
For DC, it's much better to use $\pm x_e$, where $x_e$ is the anticipated error bars, or effective number of bits, or something.  There's plenty of examples out there -- I'd take my guidance from voltmeter specs, or from industrial instrument specs.
* You are adding to precision, not accuracy.  Accuracy is a different thing**.  The accuracy of an ADC is limited by things other than its noise characteristics.
** The best short description I have is throwing darts: accuracy is how well-centered the darts are around the bullseye -- precision is how tightly clumped they are.  If all your darts land on a point 3&quot; over and 1&quot; down from the bullseye, you're very precise -- but not terribly accurate.