I'm largely a self-taught highschooler in basic Calculus and I'm utterly confused regarding what Indefinite integrals (or antiderivatives) do mean geometrically (if they really do), physically or mathematically at all (in the intuitive level). My exact confusion is what relation do they have with $x$? For example, the value of the derivative at $x$ is the slope (physical meaning) of the tangent at $x$ similarly, what relation does the value of the antiderivative at $x$ have with it?
Do Indefinite integrals have anything to do with area under the graph and if yes, from where to where?
Also how does the constant add to the geometrical significance of the antiderivatives?
Jair is right - keep reading and it will become clear. But you probably knew that =)
To tide you over temporarily: consider some continuous function $f(x)$, like a polynomial. And choose one with a bunch of distinct roots, like a $5$th or $10$th degree polynomial so it looks like a wiggly earthworm over some interval. At this point you know $f'(x)$ has the interpretation of the slope of the tangent at $x$. But $f'(x)$ is also just a function you could plot - another wiggly plot - forgetting that it represents the slopes of its parent function $f(x)$. And you can repeat this process until no more derivatives are possible ($n$ derivatives for an $n$-th order polynomial). So you can generate a linked set of functions, each of which describes the tangents of its parent function, and each of which can also be described by its own derivative.
Now just imagine that process in reverse. Knowing that a function $f'(x)$ describes the tangents of some parent $f(x)$, you can see that the "shape" of the parent function is fixed. But also, the slopes remain the same if the parent function is shifted up or down, so there is an infinite family of possible parent $f(x)$ functions, all with identical shapes but displaced vertically by some constant (which disappears on differentiation). This is the reason for the constant of integration in the indefinite integral.
The next step, which your book will take for you, is to apply those slopes to tiny trapezoids or rectangles built under the parent function, which yield the area interpretation for the definite integral.
For some applications, you will need to figure out the constant of integration (the vertical shift of the parent function) to solve some problem, but in many applications the constant will cancel itself out, your book will have plenty to say about this too.