Why does time speed up when away from gravitational masses? I read something about if an astronaut travels away from earth and comes back he/she is older because time passes faster while away from so much gravity. Would that also mean time would go slower while on Jupiter because it has more gravity than earth?
 A: I can possibly answer the why part of your question, based on these assumptions below. Your question about gravity is at the end of this attempt at an answer.


*

*The fastest velocity we know of is the speed of light. 

*We know, because we have tested it,  that we live in a universe where travel in time and travel in space are linked by simple equations. I don't mean anything exotic when I say travel in time, I just mean that time passes for us in the normal way, it's 1 pm, 5 hours later it's 6 pm. 
No big deal.

*We know that we cannot travel ourselves at the speed of light through space, think just space for the moment.

*But although the idea of travelling through time is just the ordinary passage of time, we have devised a system that measures how  fast we travel through spacetime (all one word, not separate time or seperate space).

*The difficult bit (and possibly the wrong bit if I have misunderstood Relativity, or I am using outdated examples), is if you can imagine that our speed through spacetime is the speed of light, then most of this speed is travel through time (which we are used to as the normal passage of time) but a small bit of this speed is the speed we travel at through space,  we do it every second as the Earth moves around the Sun.

*So if everybody moves through spacetime at the speed of light, then if we move faster through space alone, as we would in a rocket, to keep us at lightspeed through spacetime, we must slow down in time to balance the books and keep our speed through spacetime at lightspeed.
Einstein had two sets of laws in the late 1800s.  He had Newton's laws, whch said the passage of time never changed for anybody, no matter what their speed and he had Maxwell's laws, which contradicted this. They said that the speed of light is the same for everybody, even if they travelled at 99% of the speed of light, if they turned on a torch the light would still travel ahead of them at the same speed as it would on Earth.
To fix this up, Einstein special theory of relativity assumed that there was no absolute time, that time passes slower depending on the speed you travelled at, which is what I write above. That is why time slows down the faster you move through space. This effect is not noticeable for you in your rocket, but people on Earth will think time is going slower for you. You will think that you are stopped and it is the Earth that is undergoing a slowing down of time, because it will seem to you to be travelling away very fast.
As there is no absolute time, neither you in the rocket, nor the people on Earth are correct or incorrect. Your time always feels normal, it's always the other guy's time that feels slower to you.
Gravity has the same effect as moving very fast as regards time. If you are near a strong gravitional field, you will feel normal, but people far away from you will feel that your time has slowed down.
A: There are two different effects that cause observers in different frames to see the other's clock running at a different rate than theirs. The first is due to Special Relativity and the two frames having a relative velocity between them.  The second is due to General Relativity and the two frames being at different gravitational potentials.  The question seems to be about the second GR effect so that's what I'll describe.
Actually, you don't need the mathematics of GR to see what is happening.  Consider some cesium atoms emitting a single frequency $\nu$ radio wave.  An adjacent counter records each electric field peak of the wave as it goes by. We call the number on the counter display, the clock time. Each peak passing is one tick of the clock.  Next we have two identical cesium clocks, A and B, each with an observer sitting beside their clock.  What follows are some experiments with results which are obvious when thought about.
1) If A and B are together, both A and B see the two clocks ticking in unison...no matter whether they are together on earth or together way out in space.
2) If A is on earth and B is way up in space then:
a) Light (photons) emitted by A loose energy as they climb out of the earth's potential well. B sees these photons as lower frequency light compared to his own clock B light.  That is, B sees clock A ticking slower than his own.
b) Light (photons) emitted by B gain energy as they fall into the earth's potential well. A sees these photons as higher frequency light compared to his own clock A light.  That is, A sees clock B ticking faster than his own.
This is the effect that A can exploit to become younger than his twin B.  Initially A and B are together in space with their counters having the same value. A quickly travels to the earth's surface.  B now sees clock A ticking slower than his own. For each tick of A, B sees A's counter increment, and B sees A's counter is falling behind his own.  This continues for some arbitrarily long time, and then A quickly returns to B out in space.  Nothing magic happens to A's counter display in the quick return to B. When they are together again, both look at their counter displays and agree that A < B.  In fact, the longer they were apart the more counter A is less than counter B.
Since observer A was always beside clock A, clock A faithfully records the aging of observer A. Since observer B was always beside clock B, clock B faithfully records the aging of observer B.  Therefore, A is younger than his twin B when they get back together.
You can repeat this argument from A's point of view and get the same result.
For the last part of your question, you are correct. The twin that was at a much deeper gravitational potential on Jupiter would be younger than the one on earth when they got back together.
A: This is actually the concept of time travel. Imagine you somehow manage to orbit a black hole with your spaceship at a very small distance without crossing the horizon. Your clock runs smoothly as on earth. You notice no difference. Then at some point (after 3 days) you get bored, because it gets really lonely out there and you decide to come back to earth (ignore the travel time) only to find out that years have passed by since you left and there is no physics.stackexchange.com to read the replies to your answer. We can all agree that you have travelled in the future, just by exploiting spacetime and its properties. Time dilation is due to curvature. The deeper you descend into a "gravity well" the slower your clock runs according to an external observer that is far away from the mass that bends spacetime there. You can actually time travel on earth. All you would have to do is:    
a) have a friend
b) buy him a high precision atomic clock
c) buy yourself a high precision atomic clock
d) rent a helicopter so that he can go to Everest
e) sit at home and do some math for 24h
f) Also fly to Everest afterwards or tell him to come by your place
Check out your clocks. According to his, you travelled some nanoseconds into the future and did so without a DeLorean, just because you were closer to the source. For space flights, time slows down due to velocity, but speeds up due to distance from the source. So, one cannot answer if time for astronauts runs slower or faster according to clocks on earth, unless he knows the vessel's velocity and the altitude of the orbit. There is no need to introduce math into this topic right now, but you could imagine the notion of gravitational time dilation as due to the stretching of a fabric (very roughly speaking), which is space and time as one. 
A: Time indeed goes slower near massive bodies, but also slower when something is moving fast. So, depending on how fast the astronaut travels, when she returns she might be younger or older. The time on the International Space Station actually passes a tiny bit faster than on Earth, because the gravitational time speedup is twice as strong as the relativistic time dilation from the ISS orbiting the Earth. 
And you are correct, time near Jupiter passes slower then on Earth.
A: This is related with Time-Dialation , yes as per general theory of relativity which is experimentally found to be true in many cases, Einstein says time is not absolute that means Time doesnt have same value for different observer when we measured at different location and at different observer speed. e.g. imagine lighting strike through ground and two observers looking at it while it strike ground but one observer is standing still while other is travelling in train so both will experience same phenomenon at different speed and different time even though observed quantity is same, that means multiple observers will agree light has travelled but time measures between any two location for same phenomenon not necessarily agree. But this will unhealthy to digest on earth as we all experience almost same time but practically it was different with microscopic range.
So when we travelled faster in space time gets slower with other observer point of view. But whats this relation of light/distance with relationship with gravity. Well light generally travels in straight line that we absolutely adopted but due to gravitational field and space-time curve light gets bend so to travel at same point it need to travel at larger distance which further slows time
And thats why in space since gravity effect was more as compared to earths gravity at jupiter or at sun time slows.
Hope that helps you. ( Forgive any errors in script/grammar)
