I have a hard time understanding what an operating point and load line represent and why do we need them at all.
Do not worry because it is not just you who have this problem. For example, I first started thinking of this concept as a student in the 70's... and I continue thinking of it even now when explaining it to my students... and even when writing this answer. Below are my insights (some of them shared in the cited Wikipedia page). Тhey can help you not only to formally know... but to truly understand what these formal terms mean...
1. Ideal source and load. The most elementary electric circuit consists only of two 2-terminal elements connected to each other - a source and a load. A typical example is a voltage source connected to a resistor (Ohm's experiment) - Fig. 1a. This humble circuit is described by a simple equation, which in this case is Ohm's law - V = I.R.
Fig. 1. An "ideal" voltage source supplying a resistor (a picture from 90's); E is the voltage V. The triangle with catheti VA and IA, and angle tan R, is an excellent geometrical interpretation of Ohm's law (more meaningfull than the so-called "Ohm's triangle").
Another (less common) example is a current source connected to a resistor (another version of Ohm's experiment) - Fig. 2a.
Fig. 2. An "ideal" current source supplying a resistor (a picture from 90's); the circle on the left is the current source. The triangle with catheti VA and IA, and angle tan R geometrically represents Ohm's law.
The idea. Each of these elements can be graphically represented in the coordinate system by its IV curve (the set of all possible pairs of voltage and current). In the first example above (Fig. 1b), the IV curve of the voltage source (the source line) is a vertical line shifted to right in the case of positive voltage; the IV curve of the resistor (the load line) is an inclined (to right) line passing through the origin of the coordinate system. Since the voltage across both elements and the current through them are the same, we can impose their IV curves on each other in the same coordinate system. The intersection point A of the two curves represents the graphical solution of the circuit equation and is named operating point. Its coordinates determine the instant values of the voltage and current of the circuit.
In the dual example (Fig. 2b), the IV curve of the current source (the source line) is a horizontal line shifted up in the case of positive current; the IV curve of the resistor (the load line) is the same as above.
Varying voltage, constant resistance (Fig. 3). This graphical representation can be used to visualize the circuit operation when we vary the voltage, resistance or even both. For example, when exploring the passive voltage-to-current converter, the voltage source IV curve moves in the horizontal direction (translates) while the resistor IV curve stays immovable:
Fig. 3. Varying the voltage as an input quantity in the Ohm's circuit (voltage-to-current converter)
Besides linear (ohmic) resistors, all sorts of non-linear elements and even sources can serve as a load (diodes, transistors, etc.). For example, this setup is used to measure the (horizontal part of) transistor output characteristic.
Constant voltage, varying resistance (Fig. 4). When exploring the passive resistance-to-current converter, the resistor IV curve rotates while the voltage source IV curve stays immovable:
Fig. 4. Varying the resistance as an input quantity in the Ohm's circuit (resistance-to-current converter)
Varying current, constant resistance (Fig. 5). In the dual configuration (a current source connected to a resistor), when exploring the passive current-to-voltage converter, the current source IV curve moves in the vertical direction (translates) while the resistor IV curve stays immovable:
Fig. 5. Varying the current as an input quantity in the dual Ohm's circuit (current-to-voltage converter)
Constant current, varying resistance (Fig. 6). When exploring the passive resistance-to-voltage converter, the resistor IV curve rotates while the current source IV curve stays immovable:
Fig. 6. Varying the resistance as an input quantity in the dual Ohm's circuit (resistance-to-voltage converter)
As above, all sorts of non-linear elements and even sources can serve as a load. For example, this setup is used to measure the (vertical part of) diode IV curve.
2. Real source and load. The graphical solution is a result of intersection between two curves in the operating point; so this technique can be applied to a circuit of two elements connected to each other. In circuits with more elements, we can reduce the complex circuit by means of equivalent transformations to an equivalent circuit consisting of two parts - a real voltage source and a load. A typical example of this technique is the voltage divider configuration (Fig. 7):
Fig. 7. The voltage divider operation geometrically represented by two intersected IV curves (a picture from 90's). E1 is the voltage V of the voltage source; UR1 and UR2 are the voltage drops VR1 and VR2 across the resistors R1 and R2.
Here, two elements in series - the "ideal" voltage source E1 and the upper resistor R1, are combined into a new "composed" element E1R1. It can be thought as a real voltage source with voltage E1 and internal resistance R1. Its IV curve is a line shifted to right by E1 and inclined to left by an angle R1. It is obtained by subtracting R1 IV curve (an inclined to right line passing through the origin of the coordinate system) from E1 IV curve (a vertical line shifted to right with E1) - Fig. 8:
Fig. 8. Revealing what the "load line" actually is...
This configuration is widely used to illustrate and roughly calculate transistor amplifying stages where the name of "load line" came from. For example, in the circuit of the common-emitter stage - Fig. 9, we can combine the collector resistor Rc and the voltage source E (the power supply VCC) into a real voltage source ERc with voltage E and internal resistance Rc.
Fig. 9. In the common-emitter stage, the network of power supply and collector resistor in series can be thought as a real voltage source with voltage E and internal resistance Rc. The transistor acts as a varying current-stable resistor.
The other element - the transistor, will act as a current-stable non-linear load. So, in this arrangement, a real voltage source with linear internal resistance drives a load with non-linear resistance. Only, the conventional viewpoint is that the transistor drives the load Rc; hence the name "load line".
3. Two real sources. If there is another voltage source, it can be combined with the load. For example, in the case of R1-R2 voltage divider, if we insert another voltage source between R2 and ground, we obtain the useful circuit of a resistor voltage summer. It is used in op-amp inverting circuits - Fig. 10.
Fig. 10. Op-amp inverting amplifier illustrated graphically (a picture from 90's). Ein (VIN) and R1 form the one element; EA (Vout) and R2 form the other element. The question is, "What is source and what load line here?"
Finally, what was the "load line"?
Simply speaking, "load line" is the IV curve of a resistor and voltage source in series.