Let me try at "Why, and when 2 ellipses are good", and since I can't picture a scenario where 3 could be remotely good for any similar reasons, I think this should suffice. Also, I'll describe raising orbits, as it's easier to "feel" - but the situation is always reversible so lowering orbits works the same.
When performing orbital maneuvers against a single body, there are two situations where you are the lord over your trajectory; spend little delta-V, modify your orbit by a lot.
- Situations where you use Oberth effect - zipping past the central body at insane speeds; that $^2$ in your kinetic energy equation squares the value of your fuel there. You are limited in what you can do, because your kinetic energy is directed: you can cheaply raise/lower apoapsis, or escape (/capture). Other maneuvers become quite expensive, but that one is very important.
- Situations where you are barely subject to gravity and inertia at all. You're far, far away, moving slowly. By performing small, lazy maneuvers you can change your trajectory to pretty much anything legal. An enormous circular orbit, over the outskirts of the system? Why not? Make it polar? Cheapo. Turn from prograde orbit to retrograde? Sure, it's like a dozen m/s cost! Since everything is so lazy and slow, it will take years, or decades, until you actually complete one of the orbits you achieved. But hey, you did achieve these fancy orbits and it cost peanuts!
As you see, down, low, and high up are your places of power. It's all these damned middlings that are a hellish limbo hard to enter and hard to escape. Not enough speed for the $^2$ of Oberth to truly kick in like a legendary turbo, still enough speed that it costs an arm and a leg in fuel to change anything meaningfully.
Hohmann transfer is a straightforward thing. Raise Ap, circularize. Nothing weird, nothing fancy, works completely fine for small changes, and when you go very far, to the distant outskirts, it still works fine, as you're going through your "two places of power". But it starts ailing for the middlings. You raise your Ap just fine, and then what? And arm and a leg in fuel to circularize; no Oberth to help; the entire 3km/s of GEO speed out of your fuel tanks at no discounts!
That's when you start wishing for a discount - and use both discounts you can have.
As you start off with the Hohmann transfer orbit, you first apply your Oberth discount at raising apoapsis, to go far, far above, on the cheap. Then you go there and using your wild liberties to change orbits as you desire, you make the Pe align with the orbit you want. And yeah, you still need to pay the circularization cost, and you've already spent extra fuel to access your 'discount maneuvers'. But hey! Now you're zipping in from the outside! And that means you're now subject to another Oberth discount! Not as large as before, because the orbit is higher and slower, but hey, you're moving faster than necessary and need to slow down, instead of accelerating from standstill where your Oberth benefits were zero. That's where we're getting our savings!
And as you see, "two happy places", you start off at one, reach the other, then go to the destination. Two transfers, two ellipses. Might be different if you were starting at one middling and going to a different middling... except then you'd be near enough that plain Hohmann would come in cheaper.
...unless you're doing tricks like orbital plane change. But that's a different tale.
