The simple answer is: Changes in resistance/impedance at the connector or cable can change the actual or perceived frequency response that you hear from the speaker.
In the case of the actual frequency response: The connector and stuff inside the speaker form an RC filter (a low-pass filter). The connector is the R and the stuff inside the speaker is the C. As R goes up, the cutoff frequency of the low-pass filter goes up. Normally, the R is less than 1 ohm, so the cutoff frequency is very low (less than 5 Hz). But if the connector is bad or dirty then the R could be several hundred ohms, which could increase the cutoff frequency by several hundred times. The net effect is that the bass is all cut out.
And for perceived frequency response: Human hearing is very weird. At low volume levels, our ears simply don't pick up some frequencies very well-- usually the low and high frequencies. We're more sensitive to mid frequencies. So, as you reduce the volume level we perceive that the low and high frequencies are being cut out more than the middle frequencies. The "Loudness" button of a stereo attempts to compensate for this by boosting those frequencies and therefore making us hear a more balanced sound at low listening levels. If the connector is bad or dirty, less power will be making it to the speaker and causing a lower volume level. But due to human hearing, we would also hear less low and high frequencies-- making it sound like we're listening through a phone or something.
This is an over-simplified answer, but reasonably accurate for your purposes.
It's not going through you.
According to your drawing, the reason this works is actually the opposite of what you intuit. The audio signal isn't going through your body. It's going through the wire.
The output from the PC audio port is AC-coupled which means that it is pushing and pulling on the mobile charges in the wire with an electric field, not direct conduction. More simply, there is a gap between the wire touching the PC's internal amplifier output and the pin of the audio jack where your wire is connected.
In order for the amplifier to receive this pushing and pulling on the charges (the audio signal), everyone (PC and Speaker's amplifier's input) has to agree on what zero means (e.g. no pushing or pulling). In physical terms, this is the point where the speaker's diaphragm is in the center (neither pushing or pulling on the air) -- where it sits without power.
To provide this "zero" (we call that the reference potential or just "reference"), the second wire is included in the cable.
When you only connect the signal wire and not the reference, the amplifier receiving it doesn't actually perceive that the signal is being pushed or pulled on because the entire amplifier moves up and down. It's like being on a perfectly smooth and flat surface while standing on a skateboard and having someone pull you on a rope. All of you translates.
But if someone takes you off the skateboard and then pulls on the rope, your feet are stuck and you fall forward (you feel the rope pulling on you relative to your feet).
In your audio case, your body contains a giant mass of mobile charges. When you do not touch the reference contact of the audio connector (the area at the back of the TRS plug closest to the plastic overmoulding) it is in contact with air which has almost no mobile charges and it's like standing on the skateboard. When you touch it, you add (electrically) your entire body worth of mobile charges. This is a stable enough reference to allow the amplifier to see the wire moving as distinct from you moving.
This effect is the basis of Capacitive Touch Screens.
My explanation is an accessible approach to understanding how capacitance in general works. A similar situation to the one you described explains how capacitive touch screens work (as are found on modern smart-phones, like the iPhone).
The screen is transmitting the equivalent of an audio signal all the time and you touching the screen adds your body's mobile charges to the picture and distorts the screen's "audio" signal. It is this distortion introduced by your finger that is detected, not your "touching" of the screen. That's why capacitive touch panels don't work well (or at all) with gloves or when your fingers are wet. These conditions change the way your finger distorts the electric field and confuses the detector.
Best Answer
Probably one of the wires coming out of the plug has broken. Cut off the plug along with a few inches of cable (to get past the point where a wire is broken), then solder on a new plug.