The need for modulation:
Your voice signals generally lies in the frequency range 1kHz to 4kHz and music lies in the
range 20Hz to 20Khz. Say your locality has 4-5 AM broadcast stations. Now all the AM broadcast stations cannot transmit their content as is because if all the stations transmit at the same time there will be signal interference. Also, for efficient radiation of electromagnetic energy the radiating antenna should be of the order of the wavelength of signal radiated. If the signal is 20Hz-20kHz the length of antenna has to be of the size of kilometers. So, they place the information carried by 20Hz-20kHz at higher frequencies. Every station places its their content in their own band of frequencies. Like one AM station will operate on 100Mhz with their band of operation being 100Mhz - 20Hz and 100Mhz+20Khz and the second one at 105Mhz with their band of operation being 105Mhz - 20Hz and 105Mhz+20Khz and so and so forth. The process of placing the frequency content from one spectrum on to another is called modulations. Now this information is radiated in to free space by an antenna at the station.
The receiver has a tuning knob, which lets you select a band of frequencies. Say, you have tuned your receiver to receive station 3 which is operating at 110 Mhz. You will select all the frequencies in the range 110Mhz - 20Hz and 110Mhz+20Khz. The signal being present in the spectrum 110Mhz - 20Hz and 110Mhz+20Khz is brought back to 20 Hz to 20kHz using a process which reverses the operation in encoder side. This is called demodulation. This signal is ready to hear.
For a simple modulation-demodulation read this
Finally, if you want to post-process the signal before listening like reducing the noise, increasing bass/treble, you have to filter the signal.
In a software defined radio, there will be a RF receiver which does the job of the receiver mentioned above. The message signal that was recovered is then converted to digital samples using an analog-to-digital converter. The analog to digital converter has 2 steps, 1: Sampling at rate above or equal to Nyquist and 2. Quantization. These digital samples are processed using a software on a pc/embedded system. You can apply filters on these samples. These digital/discrete samples is then converted to analog using a D/A Converter. Main step of a D/A converter is interpolation.
Theoretically, in an ideal system, it is sufficient to sample the signal at Nyquist rate. Before play back, an ideal low pass filter will recover the message/baseband signal with out any information loss. But such a design is not feasible.
At a low sampling rate, the interpolation filter (low-pass filter) will not be able to interpolate intermediate samples effectively. However if the sampling rate is increased you can have better reconstructed message signal. Hence at higher msps you must receive a better quality signal than at low msps. However, the quality you perceive will become negligible after certain point as your ear might not be able to negotiate the difference.
I believe lowering the msps below Nyquist can introduce cross talk. That doesn't mean that you will receive neighbor radio stations because they got clipped at the RF receiver stage. The cross talk you receive is because of frequency warping.
In the above description i am speculating that sampling happens on the demodulated signal not on the modulated signal. Because demodulation can be easily achieved via an envelope detector for AM or a phase locked loop for FM. Demodulation in digital domain can be costly as sampling a high frequency signal at nyquist rate and operating at such high bandwidths can be resource consuming.
Also no FM station operates at Ghz. Giga is for satellite. 100 Mhz for FM. and Khz for AM.