I didn't see this when you first posted it but I have three standard mental tools to understand why "invariant" means "maximum possible speed" that I wanted to share with you.
Race a light pulse
Suppose you want to race a light pulse and beat it, proving that it's not the maximum possible speed. We set up a track for you to fly your spaceship on, out in space, and I scatter some sparse dust uniformly next to that track, for the laser pulse. (The dust will reflect little bits of the light so that we can see where the heck it is, the laser pulse will however be very strong so that this does not attenuate it too far beyond detection.)
What follows is actually a real-life Zeno paradox. Let's suppose that right after we start, you accelerate to $c/2$ relative to me, and try to figure out how fast this light is moving away from you. What answer will you get? Well, if it's truly invariant, you'll find that it's still moving away from you at speed $c$. Frustrated, you drop a reflector and accelerate to $c/2$ relative to that reflector, and see if you've caught up yet: no, it's still moving away from you at speed $c$. So you drop another reflector and accelerate to $c/2$ relative to that. It's still moving away at $c$: you cannot win! Unlike with Zeno's real paradoxes, the "distance you need to travel" (really, the velocity change you need to effect by accelerating) is not actually decreasing as you travel halfway to where you're going.
Just call her, already!
The above technically proves that continuous accelerations can't go faster than light but we might be more interested in the idea that information can't be broadcast faster than light. To easily understand this, consider two consequences of relativity: (1) that everyone has valid laws of physics related to moving reference frames by these Lorentz transformations, and (2) that these Lorentz transformations predict that moving clocks run in slow motion.
Alice is in a spaceship moving relative to Bob, so Bob sees Alice's clocks ticking in slow motion. But Alice also sees Bob's clocks ticking in slow motion. This is a frustrating situation! You want to say: whose clocks really are ticking slow, here?! I like to imagine that I get frustrated with this situation and say to Bob, "Just call her up and one of you will be talking fast and one of you will be talking slow and we will both know which one of you is talking slowly!"
Well, not so fast. My intuition above is treating phone conversations more or less the way they work in my conventional experience with my friends, where the communication is instantaneous between us both. But how is Bob going to call Alice up--with a cell phone? How do those work? Microwaves, which are light waves with a wavelength about the size of a hand or so. So those bits of conversation transfer at the speed of light! But that means that between whenever Bob says something and when Alice gets it, there will be a transmission time gap between those two events. That time gap will swallow up any ability to detect who is talking slow.
So we have plainly seen that instantaneous communication breaks the equivalence postulate, so some limit of faster-than-light communication in general probably lets us figure out objectively who is talking slow vs. who is talking fast. But we can preserve this equivalence of all reference frames simply by stating that no information moves faster than $c$.
Bubbles and time-travel.
I've talked a few times about expanding light bubbles on this site,
e.g. here, it's just a way of talking about what in relativity is more formally called a "future-pointing light cone." The idea is that when some sudden event happens, light rushes out in all directions at speed $c$ to notify everyone about this event that's happened: that structure of expanding-at-$c$ bubbles of light (they're thin because the events are instantaneous) is one way of thinking about what relativity is all about.
The Lorentz transforms that relativity allows maps all expanding bubbles to other expanding bubbles, but it might grow or shrink different bubbles differently. However, Lorentz transforms will always respect the bubbles' topology: if one light-bubble is contained within another light-bubble then if I shrink the outer one I cannot grow the inner one until they collide; the one must remain topologically inside the other one. Similarly if two expanding bubbles intersect on a circle, I cannot do anything to put one of them wholly inside the other one; as they shrink smaller and smaller they must become disconnected bubbles.
The first topology is called a "timelike separation" of the two events; the second topology is called a "spacelike separation" of the two events. (There is one other option, where one of them is inside the other one but they both share exactly one point on the surface of the sphere, and this is called a "null separation" of the two events, it is just on the border between these, where if the inner one were just a little bigger they'd intersect on a circle and if it were just a little smaller they'd not intersect at all.)
Importantly, if two events are timelike-separated then there is no objective space separation; there is some allowed Lorentz transform such that both light bubbles are centered on the same point and these reference frames think that both events happened at the same place. Similarly if they are spacelike-separated then there is no objective time separation; there is a reference frame which scales both bubbles to the same size and therefore thinks that in the past they both were shrunk down to points at the same time.
Once you can appreciate these you can see that if you can do arbitrary ad-hoc faster-than-light travel as well as arbitrary slower-than-light travel, you can travel backwards in time too.
It is very simple: Consider some event in your past, you are stuck in this expanding bubble of light. Well, you're not stuck: you can travel faster than light to eventually break through the bubble and get outside of it. Now do something outside that bubble, and you'll find yourself in a new bubble, one that is spacelike-separated from the other bubble. Boosting into one of the other normal reference frames, you can shrink the original event from your past down to a point while the bubble you're in grows very large. If you do it right, you can now travel faster-than-light out of your bubble and set up a new bubble which will contain that bubble from the event in your past, proving that you have time-traveled with just two faster-than-light jumps in different directions. Evidently your starting reference frame must view this second jump as a timetravel backwards in time; and in fact it must be a theorem that all faster-than-light travel of information looks to someone like the information travels backwards in time, though you do need to be able to travel superluminally in two different reference frames in order for this to permit provable time-travel.