The reason why plots are universally used to introduce simple regression - a response predicted by a single predictor - is that they aid understanding.
However, I believe I can give something of the flavor that might aid in understanding what's going on. In this I'll mostly focus on trying to convey some of the understanding they give, which may help with some of the other aspects you'll typically encounter in reading about regression. So this answer will mainly deal with a particular aspect of your post.
Imagine you are seated before a large rectangular table such as a plain office desk, one a full arm-span long (perhaps 1.8 meters), by perhaps half that wide.
You are seated before the table in the usual position, in the middle of one long side. On this table a large number of nails (with fairly smooth heads) have been hammered into the top surface such that each pokes up a little way (enough to feel where they are, and enough to tie a string to them or attach a rubber band).
These nails are at varying distances from your edge of the desk, in such a way that toward one end (say the left end) they typically are closer to your edge of the desk and then as you move toward the other end the nail-heads tend to be further away from your edge.
Further imagine that it would be useful to have a sense of how far on average the nails are from your edge at any give position along your edge.
Choose some place along your edge of the desk and place your hand there, then reach forward directly across the table, gently dragging your hand directly back toward you, then away again, moving your hand back and forth over the nail heads. You encounter several dozen bumps from these nails - the ones within that narrow breadth of your hand (as it moves directly away from your edge, at constant distance from the left end of the desk), a section, or strip, roughly ten centimeters wide.
The idea is to figure out some average distance to a nail from your edge of the desk in that small section. Intuitively it's just the middle of the bumps we hit but if we measured each distance-to-a-nail in that hand-breadth-wide section of desk, we could compute those averages easily.
For example, we could make use of a T-square whose head slides along the edge of the desk and whose shaft runs toward the other side of the desk, but just above the desk so we don't hit the nails as it slides left or right - as we pass a given nail we can get its distance along the shaft of the T-square.
So at a progression of places along our edge we repeat this exercise of finding all the nails in a hand-width strip running toward and away from us and finding their average distance away. Perhaps we divide the desk up into hand-width strips along our edge (so every nail is encountered in exactly one strip).
Now imagine there were say 21 such strips, the first at the left edge and the last at the right edge. The means get further away from our desk-edge as we progress across the strips.
These means form a simple nonparametric regression estimator of the expectation of y (our distance-away) given x (distance along our edge from the left end), that is, E(y|x). Specifically, this is a binned nonparametric regression estimator, also called a regressogram
If those strip means increased regularly - that is, the mean was typically increasing by about the same amount-per-strip as we moved across the strips - then we could better estimate our regression function by assuming that the expected value of y was a linear function of x - i.e. that the expected value of y given x was a constant plus a multiple of x. Here the constant represents where the nails tend to be when we at x is zero (often we might place this at the extreme left edge but it doesn't have to be), and the particular multiple of x being how fast on average the mean changes as we move by one centimeter (say) to the right.
But how to find such a linear function?
Imagine that we loop one rubber band over each nail-head, and attach each to a long thin stick that lays just above the desk, on top of the nails, so that it lays somewhere near the "middle" of each strip we had be for.
We attach the bands in such a way that they only stretch in the direction toward and away from us (not left or right) - left to themselves they would pull so as to make their direction of stretch at a right-angle with the stick, but here we prevent that, so that their direction of stretch remains only in the directions toward or away from our edge of the desk. Now we let the stick settle as the bands pull it toward each nail, with more distant nails (with more stretched rubber bands) pulling correspondingly harder than nails close to the stick.
Then the combined result of all the bands pulling on the stick would be (ideally, at least) to pull the stick to minimize the sum of squared lengths of the stretched rubber bands; in that direction directly across the table the distance from our edge of the table to the stick at any given x position would be our estimate of the expected value of y given x.
This is essentially a linear regression estimate.
Now, imagine that instead of nails, we have many fruits (like small apples perhaps) hanging from a large tree and we wish to find the average distance of fruits above the ground as it varies with position on the ground. Imagine that in this case the heights above the ground get larger as we go forward and slightly larger as we move right, again in a regular fashion, so each step forward typically changes the mean height by about the same amount, and each step to the right will also change the mean by a roughly constant amount (but this stepping-right amount of change in mean is different to the stepping-forward amount of change).
If we minimize the sum of squared vertical distances from the fruits to a thin flat sheet (perhaps a thin sheet of very stiff plastic) in order to figure out how the mean height changes as we move forward or step to the right, that would be a linear regression with two predictors - a multiple regression.
These are the only two cases that plots can help understand (they can show rapidly what I just described at length, but hopefully you know have a basis on which to conceptualize the same ideas). Beyond those simplest two cases, we're left with the mathematics only.
Now take your house price example; you can represent every house's area by a distance along your edge of the desk - represent the largest house size as a position near the right edge, every other house size will be some position further to the left where a certain number of centimeters will represent some number of square meters. Now the distance away represents sale price. Represent the most expensive house as some particular distance near the furthest edge of the desk (as always, the edge furthest from your chair), and every centimeter shifted away will represent some number of Rials.
For the present imagine that we chose the representation so that the left edge of the desk corresponds to a house area of zero and the near edge to a house price of 0. We then put in a nail for each house.
We probably won't have any nails near the left end of our edge (they might be mostly toward the right and away from us) because this isn't necessarily a good choice of scale but your choice of a no-intercept model makes this a better way to discuss it.
Now in your model you force the stick to pass through a loop of string at the left corner of the near edge of the desk - thus forcing the fitted model to have price zero for area zero, which might seem natural - but imagine if there are some fairly constant components of price which affected every sale. Then it would make sense to have the intercept different from zero.
In any case, with the addition of that loop, the same rubber-band exercise as before will find our least squares estimate of the line.