There are two sides to this story:
(1) what $I$ and $Q$ are from the perspective of a user and
(2) how they are produced.
For this answer I'm going to assume familiarity with complex numbers because this is fundamentally the point of $I$ and $Q$.
Why I and Q?
Euler's formula states that
$$
\exp(i\omega t)=\cos(\omega t)+i\sin(\omega t)
$$
We can flip this around to get
$$\cos(\omega t)=\frac{1}{2}(\exp(i\omega t)+\exp(-i\omega t))$$
and
$$\sin(\omega t)=\frac{1}{2i}(\exp(i\omega t)-\exp(-i\omega t))$$
We can think of this as saying that both the $\sin$ and $\cos$ functions are a sum of two parts: one with positive frequency and one with negative frequency.
Why should we care about this? Let’s say you want to shift the frequency of your signal down by some amount, say $\omega_0$.
Suppose by some kind of magic our incoming signal was actually complex valued: $\exp(i\omega t)$. Then it's really easy to shift the frequency down - we make an oscillator that produces the signal $\exp(-i\omega_0 t)$ and multiply our incoming signal by this. Using $\exp(a)\exp(b)=\exp(a+b)$ we get $\exp(i(\omega-\omega_0)t)$.
In reality though (no pun intended) the analogue signal is real-valued and oscillators produce real-valued signals. So an incoming signal is actually more like $\cos(\omega t)$ and our local oscillator is something like $\cos(\omega_0 t)$. Multiplying these gives:
$$\cos(\omega t)\cos(\omega_0 t)=\frac{1}{2}\cos((\omega+\omega_0)t)+\frac{1}{2}\cos((\omega-\omega_0)t)
$$
So multiplying by a local oscillator does give a signal with the frequency shifted down, but sadly it has also been mixed with another signal with the frequency shifted up. In principle we can filter out the high frequency part leaving just the low frequency part but filtering is more complicated than multiplication. The reason we get this mixture of both sums and differences of frequencies is that the $\sin$ and $\cos$ functions are mixtures of positive and negative frequency parts. Multiplying by $\cos$ ends up shifting both up and down and mixing the results. If we just had the positive frequency part then shifting the frequency would be no harder than multiplication. In general, any kind of analysis of the signal will be confused by the fact that there are positive and negative frequency parts present - eg. when you want to find the instantaneous frequency in FM demodulation.
Notice how even though the original signal is real (eg. $\cos(\omega t)$), when you remove the negative frequency part you get a complex number (eg. $\exp(i\omega t)$). This complex number has real and imaginary parts. They are conventionally called $I$ and $Q$.
And that's basically it. If your incoming signal was a sum of lots of sine waves of differing amplitude and phase, ie.
$$
\sum_{k=1}^N A_k\cos(\omega_k t+\phi_k)
$$
then the signal delivered to the SDR user is
$$
I+iQ = \sum_{k=1}^N A_k\exp(i(\omega_k t+\phi_k))
$$
In practice it's conveniently shifted down in frequency for you by some amount $\omega_0$. But that's just multiplication by $\exp(-i\omega_0 t)$. So the actual $Q$ and $I$ you receive are given by
$$
I+iQ = \sum_{k=1}^N A_k\exp(i((\omega_k-\omega_0)t+\phi_k))
$$
It's now easier for users to shift the frequencies up and down to look at different parts of the spectrum. It's also convenient in other ways - for example in (vanilla) FM demodulation you can read off the signal (ie. the instantaneous frequency) as the rate of change of the phase of the complex signal. This is a tiny bit of code. Additionally I find it much easier to reason about $\exp$. There are no trig formulae to remember. Lastly, we get two samples at each moment in time which gives the potential to pack more information into our signal.
How do you generate I and Q?
The other side of the story is how $I$ and $Q$ are generated. The question doesn't explicitly ask for this but it seems to be standard practice to answer it anyway.
At first it may even seem impossible to do this. No matter how long you spend looking at the real part of a number you can't magically infer what the imaginary part is. The key is to look at filtered signals.
I mentioned that you can shift frequency down by multiplying (in the analogue hardware) by a real-valued locally generated $\cos$ signal and filtering before sampling. That's the $I$ signal. (Or at least that's one way to get it.) $Q$ is then generated by multiplying the original signal by $\sin$ instead of $\cos$, filtering and sampling.
Repeating what I said above, if the incoming signal is $\cos(\omega t)$ and we multiply by $\cos(\omega_0 t)$ we get
$$\cos(\omega t)\cos(\omega_0 t) = \frac{1}{2}\cos((\omega+\omega_0)t)+\frac{1}{2}\cos((\omega-\omega_0)t)
$$
which, when filtered, becomes
$$
\frac{1}{2}\cos((\omega-\omega_0)t)
$$
Similarly we can separately multiply the signal by $\sin(-\omega_0t)$
$$\cos(\omega t)\sin(-\omega_0 t) = \frac{1}{2}\sin((\omega-\omega_0)t)-\frac{1}{2}\sin(\omega+\omega_0)t)
$$
and filtering out the high frequencies gives
$$
\frac{1}{2}\sin((\omega-\omega_0)t)
$$.
So (apart from an irrelevant a factor of 2) this process gives us the real and imaginary parts of $\exp(i(\omega-\omega_0)t)$ which is the same as $\cos((\omega-\omega_0)t)$ with the negative frequency part removed. (You can think of this as being a way to implement multiplication by $\exp(-i\omega t)$ followed by filtering in a circuit.
I originally said that it's a pain to multiply by $\cos$ and then filter, it's easier to work with complex values. And yet here I've said that the hardware does this anyway. But there is a point to this - once you've computed $I$ and $Q$ once you can now shift the frequency up and down as often as you like by multiplication. If you only gave the user $I$ you'd have to do the multiplication and filtering again every time you tried to shift frequency.
TBH I find the standard notation a bit confusing. I think it's easiest to think of $I+iQ$ as a single entity, a complex-valued signal, rather than as two separate things with different names (“in-phase” and “quadrature”). When you dig into SDR software it often goes straight to $I+iQ$ and from that point onwards the code works with this complex representation. For example look at the demodulators for SDR++. Also, I find it confusing that many explanations start with how $I$ and $Q$ are generated. This is good if you make SDR devices, but not if you’re a user.
Note that $Q$ and $I$ are sampled simultaneously. As far as I know $Q$ is not simply another sample taken $\frac{1}{4}$ of a cycle later even though it seems popular to say things like this. What's happening is that we multiply our signal by both $\cos(\omega_0 t)$ and by $\sin(-\omega_0 t)$ and the latter is a $\frac{1}{4}$ cycle out of phase with the former. (But I guess it's possible that alternative schemes are sometimes implemented - for example instead of multiplying by $\cos(\omega_0 t)$ and by $\sin(-\omega_0 t)$ you could multiply both the original signal and a delayed signal by $\cos(\omega_0 t)$ or some variation.)
