What is the relation between tensor calculus and Einstein's field equations? or What is the contribution of tensor calculus to Einstein’s field equations?
Well many people have wrote entire books to answer this question but I will attempt to give you some high-level 'Ten-Thousand Foot View'. Hopefully by reading this you can gain a bit of context that can serve as a launching point into further investigations of your own! :)
First and foremost I would start by addressing what it is that the Einstein Field Equations are intended to provide you with as mathematical tool. By no means is this a rigorous definition, but the most basic purpose of Einstein's Field Equations are to provide the ability of describing space-time which has intrinsic curvature. This warping of space-time corresponds to what we experience as gravity.
So, now that we have a basic description of what the field equations do, we can began to explore your actual question!
"How is tensor calculus applied to Einstein's field equations?"
So to best understand the correlation between Tensor Calculus and the Field Equations, I would begin to think about the following. In school when you are learning basic geometry and trigonometry, you are working within a coordinate system which is referred to as the Cartesian Frame. In other words, this is the standard grid system where you have an intersecting horizontal and vertical line for each integer location forming four quadrants in a 2D plane. Now, when you first learn calculus, you are learning how to deal with curves and motion within a cartesian coordinate system.
Here is where Tensor Calculus comes into play! The Einstein Field Equations are generalized to work irrespective of the coordinate system which you define. In order to accomplish this, the field equations utilize Tensor Calculus, which has the same relationships as mentioned in the previous paragraphs, but with the added capability of transforming between coordinate systems arbitrarily.
So in conclusion, the same way that if you wish to describe the motion of a 3D object when thrown through the air, utilizing basic calculus and trig in a cartesian coordinate system, you would use Tensor Calculus via the Einstein Field Equations to describe that objects motion IRRESPECTIVE of the underlying coordinate system, while also describing the effect which the object itself has on the intrinsic curvature of the space which it travels through. (The path that an object will take is called a Geodesic for further research if you so desire!)
Hopefully this helps clarify a bit, that the Field Equations are defined USING tensor calculus in order to grant the extra properties discussed in the above sections. There are many things that I have left out and like I said whole books are written about this topic, but this is what I believe is the core of what you are trying to discern! Let me know if you need anything further clarified!