What can be done with I/Q data which would otherwise be impossible?
The main motivation for using quadrature (I/Q) signals is spectral efficiency.
Most signals are naturally baseband: they have no carrier, their energy is concentrated around DC, and they are limited to bandwidth $B$. An example would be an audio signal, which has bandwidth of roughly 20 kilohertz.
To transmit such a signal, often it must be modulated (or &quot;upconverted&quot;) using a carrier. For example, in AM broadcasting, the audio signal is first limited to a bandwidth $B=10\,\text{kHz}$, and then upconverted to a frequency of roughly one megahertz.
The issue is that, when upconverted, a signal occupies a bandwidth $2B$, which is double the original bandwidth $B$. So, an AM signal occupies $20\,\text{kHz}$ &quot;on the air&quot;. The extra 10 kilohertz do not have any additional information; they are simply a loss of half of the available bandwidth.
Since bandwidth is a very limited and expensive resource, such a waste is unacceptable.
One way around this issue is to use single side-band modulation (SSB), which eliminates the waste. However, SSB signals are more difficult to generate and have other disadvantages.
The most popular solution is to use quadrature signals. This allows transmitting two signals of bandwidth $B$ over a chanel of width $2B$, restoring the average spectral occupancy to $B$ hertz per signal.
To see more of the math behind quadrature transmitters and receivers, see this answer.