Why does a longer fiber optic cable result in lower attenuation?

This is where the measurement scientist has to go into full sceptical and investigative mode.

First thing. Fibre, as a passive material, is lossy. It absorbs power. Therefore the power arriving at the end of a length of fibre will be less than was launched. Period. No arguments. We don't do over-unity here.

So what causes your observations?

Single mode, 1m -36.14dBm, 10m -36.12dBm

How repeatable are your measurements? Break down and rebuild the connections, and measure again, several times (min 3, but 5 or 10 would be better). Only then can you see whether 0.02dBm is a significant physical effect or whether it's a lucky coincidence.

Measure 20m, and 30m. Is 0dB +/- 0.1dB a reasonable absorption level for 10m of fibre? I don't know, that's what you are measuring. You can be assured that the fibre loss in dB will be additive for longer lengths (for single mode, if there are multiple modes propagating this may not be true for the total power, but it's still true for each mode), so (once you're in single mode operation) you should be able to draw a linear graph of fibre length against dB loss. Remember, 2 points makes a very statistically poor graph.

And finally, I used the phrases 'arriving at the end' and 'the power that was launched'. The power in the fibre isn't necessarily the same as in the test gear. The interfaces will create uncertainty, they lose power. The power losses depend on axial alignment, the gap, the fibre face surface finish (so how well it was prepared). I would be completely unfazed by a measurement showing that a short length of fibre had a lower loss than just the source directly into the receiver, because it's about optical coupling efficiency.

Further to the repeatability measurements I asked you to make above, that's not just several repeat assemblings of the same components (which is measuring your variability), but also doing it again for different samples of nominally the same components (the variability of the system and whether the tools and methods you are provided with work repeatably). So make 3 or more samples of 1m fibre, and compare them.

Single mode 1m 36.14dBm, multimode 1m 35.94dBm

Again, characterise your repeatablity, before you jump to any conclusions on whether a measured difference of 0.2dB is significant.

Single and multi mode fibres might have different optical apertures, so have different coupling losses, quite independent of their transmission losses. Prepare some 'zero length' fibres, or as near zero as the apparatus allows, and measure those. And do 10m, 20m, 30m plots for both. Then you can start saying that there is a significant difference between them.

Multimode 1m -35.94, 10m -18.48dBm

No. Given your other measurements above, something's wrong. You've spilt coffee on the apparatus, or someone's adjusted something while your back was turned, for a laugh. Measure again.

So you thought making measurements and drawing conclusions was easy? No. Test any difference you see against your experimental repeatability. Vary one factor at a time. Consider all possible factors and control for them all. Remember, if a difference is real, it will persist as you make repeated measurements. If you just see something one time, is it the effect, is it you, is it something you hadn't thought of?

The other answers have suggested some ways your experiment might have gone wrong. Let me tell you how to do a fiber attenuation measurement correctly.

The standard technique is called a cut-back measurement.

This means you set up your source feeding a long piece of fiber (say, 10 m). You then direct the output of that fiber into a large-area detector (large enough that it captures essentially all the light exiting the fiber), or into an integrating sphere (which is really the best way to capture all the output light). Measure the light output.

Now, without disturbing how the light is coupled in to the fiber, cut the fiber back to a shorter length (1 m in your case). Capture the output light the same way you did before, and measure the output power.

The reason to use this technique is that the launch efficiency is usually highly variable, particularly in benchtop measurements. You could easily add or subtract 3 or 6 dB (or much more, for single-mode fiber) just by misaligning the fiber to the light source by a fraction of a degree or a few microns of position. This is likely one source of error in your experiment, although you didn't describe how or when you disconnected and reconnected the source.

Another issue to watch out for is cladding modes. This is light that is coupled in to the cladding and may propagate for a few meters, but will experience higher attenuation than light in the desired modes. To avoid measuring cladding mode effects, it would be better to use longer fiber lengths for your measurement. For example, start with 100 m of fiber, and cut it back to 90 m to do the attenuation measurement.

Edit: One more issue. If you are measuring such short lengths, you'll need to be sure your light source is incredibly stable. Probably first measure the light source every second for a few hours to make sure its output power doesn't vary by more than a small fraction of the attenuation you expect from your fiber.

Neil_UK's answer is pretty much spot on, i.e. your measurements are broken. :-(

The first and most obvious problem is in the lengths chosen, 1m and 30m: These are both well within the edge effect ranges, i.e. the quality of the fiber end connections will dominate any actual attenuation loss.

In particular, good quality single mode fiber at 1300 nm can come very close to the theoretical minimum loss which is a small fraction of a dB per km, this is how transatlantic cables can work with just a few amplifiers along the way.

If we assume cheaper fiber in the 0.1 to 1 dB/km range, the 30m length still gives negligible loss. Please try 1-10 km!