Most LED strips are constructed so that you can cut them at periodic intervals so you can customize the length. What this means is that the LEDs are arranged in a parallel-series configuration. For instance, you might have a 24V strip that can be cut at every 12 LEDs. So if you have a strip of 48 LEDs it is electrically the same as four strips of 12 LEDs in series and the four strips are arranged in parallel with each other.
Now suppose you have N of these 48 LED sections and you join them end-to-end. Although it looks like you are creating a series combination of the strips, you are really joining 4*N strips of 12-LED segments each in parallel to each other.
The upshot is that connecting these kinds of strips "in series" with each other is the same as connecting them in parallel.
As far as power is concerned, it is basically proportional to the total number of LEDs. So if 1 meter requires 6.8W you should expect 6 meters to require 6*6.8 = 40.8W.
Note that you can inject power anywhere in the strip - not just at the ends. In fact, for long strips it is advisable to inject power in the middle or even at multiple points. This will reduce power loss due to the resistance in the traces that carry power to the LEDs.
Now suppose you have a 2 meter strip of LEDs. If you inject power in the middle it's the same as putting two 1-meter strips in parallel with each other. So that's another way to see that connecting segments end-to-end is the same as connecting them in parallel.