According to Einstein, mass curves spacetime and objects in the nearby field tends to travel in the shortest possible path to reach their heavier counterparts. My question is was not Newton's interpretation better; i.e. considering gravity as a force that acts on $n$ masses and tends to attract? I know that photons don't hold up with this definition, but how can spacetime be visualized (saw those trampoline models) in real life? How to define motions of celestial objects by considering an invisible 'fabric'? Any help will be appreciated.
The trampoline models do not show spacetime; they show space at one instant of time. To be precise, they offer an attempt to visualize space by taking a 2-dimensional cross-section (e.g. the equatorial plane) and then showing how spatial distances are affected by gravity by plotting a surface such that the distances along the surface match the distances in the cross-section through the gravity-affected space.
This visualization of space does offer some good intuition, but unfortunately it is not much use at understanding the idea of a geodesic or 'straightest possible line' in the temporal direction. For that you need a diagram showing time as well, and such a diagram is not so easy to draw. What I think people working in this area do is use the spatial diagram to get a feel for the notion of a spatial geodesic (the shortest spatial line between two points at some given time) and then mostly trust the algebra when they calculate timelike geodesics. These are the lines in the temporal direction that show how things move when they are moving solely under gravity.
To get an intuition about these timelike geodesics, picture the spatial diagram but flatten it out, without forgetting that the distances are distorted really, and then allow the vertical direction to represent time. A timelike geodesic extends upwards and turns towards the central axis. For a circular orbit it would be a helix. Imagine lots of little tick marks on this line, representing the ticking of a clock moving along it. If you fix the two ends of this line and then pull the middle of the line outward a little, there are fewer clock ticks along it because the clock has to move faster along the line and it gets a time dilation associated with this motion. If you push the middle of the line inwards a little, so that the clock takes a shortcut to its destination, then it can move more slowly, but now there is a gravitational time dilation that makes it tick slower on average. The line actually followed by the falling clock is the one which makes a compromise between these two effects and thus has the highest number of clock ticks between the given start and end events.
So there is one attempt at visualization. I am aware that it all seems rather abstract but in the end of course we have to go with the theory that matches experimental observation. But in this case there is also an added feature: it is the feeling that the theory has an extraordinary beauty in and of itself. The very fact that we do not need to mention the concept of force is itself to do with the fact that we can consider the whole description in geometric terms. If one day you get to study this more fully then you will be able to appreciate the beauty more fully.
If you ask "why? why does the falling clock follow the line with maximum proper time?" then one way to answer is to focus on each tiny segment of the line. The answer is that each tiny segment just goes straight ahead! But how can lots of straight segments add up to a curved line? For that your best answer is to think about beetles walking around on the surface of a sphere. A beetle walks in a "straight" line when the legs on the two sides of its body move through the same distance. But two beetles setting off from the south pole of a sphere in two different direction, and walking "in a straight line" like this, will find that their lines meet up again at the north pole. This illustrates the notion that a sequence of segments that do not themselves turn to the right nor to the left nevertheless make up a non-trivial overall line if the space (or the spacetime) is itself warped or curved.
Actually they tend to follow the longest timelike curve (a timelike geodesic).
Why is this model better than the Newtonian one? Just a reason, but there any many further. When you project onto the spatial section the geodesic of Mercury, the equation of Newton would give rise to an ellipse, whereas Einstein's geodesical equations show that the major axis of that ellipse also rotates around the sun. Astronomical observations strongly and quantitatively agree with this prediction ruling out Newton's model.
General Relativity tells us that gravity is the result of spacetime curvature, and that gravitational mass is identical to inertial mass.
In contrast, Newton did not attempt to explain why gravitational attraction occurs, he just gave a way to mathematically model that attraction. And he had no good explanation for why gravitational mass is equivalent to inertial mass. Also, he was quite unhappy that gravity acted at a distance through apparently empty space.
Famously, he used the Latin phrase "Hypotheses non fingo", meaning "I feign no hypotheses", "I frame no hypotheses", or "I contrive no hypotheses".
Here's a modern translation of that passage (from an essay appended to his Philosophiæ Naturalis Principia Mathematica), courtesy of Wikipedia:
I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction.
Admittedly, General Relativity doesn't tell us why stress-energy-momentum induces spacetime curvature, perhaps a future Quantum Gravity theory will have something to say on that score. But General Relativity is not only more mathematically accurate than Newtonian gravity, it has substantially reduced those conceptual gaps in Newton's theory.
I should mention that it's certainly much harder to do calculations of celestial motions in General Relativity than using Newton's equations. We only use GR when we need that extra accuracy, and in scenarios where Newtonian gravity is inadequate, eg, when dealing with black holes or neutron stars. And even then, we use Newtonian gravity to calculate a first approximation and then apply the necessary corrections to get the relativistic solution.
@Eisenstein just as a supplement to the previous answers, no physics theory attempts to lay bare the ultimate reality of nature, rather, all theories are mere 'models'. Physicists always favour that theory which explains experimental observations/measurements better than other candidate models. So is it with Newtonian gravity and Einstein's relativity - both give reasonably satisfactory results in their respective domains. check out this link for more proof on why we feel Einstein's theory is right
You ask if Newton's theory isn't "better". But you have to define "better" first.
There are mostly a few different requirements for good theories:
A theory could be "better" only in a specific way.
General Relativity is the most accurate theory of gravity in an absolute sense, at all scales it is expected to yield more accurate results, and it starts to differ significantly from Newtonian mechanics near massive bodies.
Newtonian gravity may be considered simpler (Euclidean geometry), although one may find the axioms of relativity more parsimonious (simpler axioms) than the Newtonian, which demands assuming the inverse square law (offering no explanation for "why" the law is as such) and gravitational and inertial mass equivalence.
Computationally, certainly Newtonian gravity is simpler (it may be interpreted as an approximation of GR), so it is the best theory for fast calculation of low mass interactions.
Metaphysically, we may equate truth with maximum possible accuracy (by definition), so finding the more accurate theories is the path to discovering the a nature of the universe, or the closest we can get. Truth may be thought to have intrinsic value (apart from the practical utilities discussed before).