SNR - signal to noise ratio - is measured at the receiver, and says how much stronger the received signal is than the underlying background noise ("noise floor"). This is limited by the amount of transmitted power, the losses on the line, how much noise is picked up on the line (crosstalk, thermal noise etc), and importantly, how fast they transmit. The faster they try to cram data bits into the same spectrum, the harder it is to detect from the noise, and hence the lower SNR.
The modems agree on what speed they will talk at, and the speed they choose will affect the SNR. Conversely, if they have decided to aim for a particular target SNR, they will adjust their speed to meet this.
For a given line, the noise floor means there is a theoretical maximum rate which can be achieved (Shannon limit) - the lower the SNR, and hence the faster you are sending, the closer you are to that limit.
A higher SNR is more "conservative": by choosing a lower data rate for the line, the signal is easier to separate from the noise at the receiver, so more reliable.
With a 6dB SNR target, the modems choose a line speed so the receiver sees the signal as 6dB "louder" than the noise. With a 3dB SNR target, they send more data in the same spectrum so the signal is only 3dB "louder" than the noise - making it harder to decode, and more susceptible to errors. Any additional noise, such as impulse noise (e.g. fridge turning off and on) is more likely to cause data corruption.
Please excuse the very woolly description, but it gives a rough idea