Taking the first part of your question first: how do we get the energy-time uncertainty from the position-momentum uncertainty? This turns out to be surprisingly difficult to do rigorously. Even Heisenberg was only able to give an approximate derivation of it (based on a property called Compton wavelength). I found some of rigorous derivations here, here and here, but these are utterly impenetrable for the beginner. Even the Compton wavelength argument is a bit involved, so what I'm going to give is a justification based on dimensional analysis. This doesn't prove the energy-time uncertainty relation, but it shows it is plausible.
The Heisenberg uncertainty principle relates position and momentum. Position has units of distance, e.g. metres, and from basic mechanics ditsance is velocity times time:
$$x = vt$$
Momentum has units of mass times velocity:
$$p = mv$$
So if you multiply together position and momentum (as the Heisenberg UP does) you get:
$$x \times p = vt \times mv = t \times mv^2$$
I've rearranged the right hand side slightly because kinetic energy is $1/2mv^2$, so the right hand side looks like time times energy i.e.
$$x \times p = t \times E$$
NB this doesn't prove that $\Delta x \Delta p = \Delta t \Delta E$ but it shows that it's plausible.
Now onto the second part of your question (assuming I've convinced you that the energy-time UP follows from the position-momentum UP).
First let's ask is the energy-time UP real. Yes it is, and we can observe it fairly easily. You've probably heard that if you excite an atom it will emit light as it returns to it's ground state, and the frequency of the light emitted depends on the energy difference between the excited and ground states. This creates the atomic spectrum, which is routinely used for identifying atoms. Helium was first identified in the atmosphere of the Sun using this technique. Anyhow, the lines in the atomic spectrum don't have a precise frequency. If you measure them carefully you'll find they span a range of frequencies. Part of the broadening is from mundane sources like the doppler shift, but part arises from the E-t uncertainty principle.
So the E-t uncertainty principle is real, and it means we can't be certain about the energy of an atom unless we watch it for an infinite time. But exactly the same argument means that if we take some patch of vacumm we can't be certain about it's energy unless we watch it for an infinite time. That means the energy of the vacuum must fluctuate i.e. energy must spring into existance from nothing.
You may still be a bit unconvinced, but we can actually measure this spontaneous creation in the vacuum using the Casimir effect, so we know it really happens.
Hopefully by now you're convinced about (temporary) creation from nothing, and I guess your next question is precisely what happens when a virtual particle is created. Sadly I can't give you an answer for this. We have mathematical models for the process, like Quantum Field Theory, but whether this is what eally happens, or even if "what really happens" is a meaningful question, I don't know.
Response to comment: this links up with my answer to your other question, What is meant by "Nothing" in Physics/Quantum Physics?, so I thought I'd expand this answer rather trying to put everything in comments.
Anyhow, you ask a fair question. I've taken the position that the vacuum is effectively nothing plus the vacuum fluctuations, and you're asking me how I know it's not something plus the vacuum fluctuations. Actually this is sort of where we came in with your first question in the series.
My answer is that we can do experiments on the vacuum to see what's there. For example we can measure the vacuum fluctuations using the Casimir effect, and we get the answer our theory predicts. We can shine light through the vacuum to see if there's anything there, and we can weigh the vacuum (i.e. see if the vacuum has any gravitational attraction). In all cases we get the results our experiment predicts, and that's why I say the vacuum is effectively nothing plus the vacuum fluctuations.
You could argue that there is something present that we haven't worked out how to detect yet, but without any theoretical backing for this it's like saying there are fairies at the end of the garden that we haven't worked out how to detect yet!