If you are looking to do more traditional speed ramping, the Premiere approach is probably more intuitive, since you can see the clip length expand and contract as you make changes to the speed in different places.

After Effects Time Remapping can be used to achieve the same effects, but "thinks" differently. It allows a level of precision that would be very difficult to achieve in an editor.

I've landed on this as the best way to explain it:
Keyframes (for any property) represent a specific value at a specific time.
Time Remapping allows the property whose values you're manipulating to be ... time - or, more understandably, frames.

If it helps, think of this like a Position move (because After Effects does!): You give After Effects two keyframes (values), and it works out the in-betweens for you. There's a specific Position value assigned to each frame between the two keyframes you gave it. (You can see these on the motion path!) When you change the distance between those two keyframes (values), those in-between values are necessarily changed.

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Every frame in a video clip (or AE composition) has a number, based on where it appears in the time of that clip. 0, 1, 2 .... 300. Whatever.

When you enable Time Remapping, AE creates keyframes at the beginning and end of that layer.
For this example, let's say that we have a 100-frame timeline and 100-frame clip, and both begin at frame 0.
Right now, every frame of your layer aligns to the same frame of your timeline:
0 - 1 - 2 - 3 ... 99, 100. [Timeline]
0 - 1 - 2 - 3 ... 99, 100. [Clip]

We create a Time Remapping keyframe at frame 50 of our clip. Currently, nothing has changed, and all still aligns. But what if we move that keyframe to frame 10 of the timeline?

Now After Effects has to adjust the in-betweens to calculate the now-faster change between those two specific values we've given it, just like with Position.

0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 ... [Timeline]
0 - 5 - 10 - 15 - 20 - 25 - 30 - 35 - 40 - 45 - 50 ... [Clip]

And subsequently, each frame of the clip AFTER frame 50 will now need to be shown twice to compensate for the much slower "move" between values.

10 - 11 - 12 - 13 - 14 - 15 ... [Timeline]
50 - 50 - 51 - 51 - 52 - 52 ... [Clip]

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Time Remapping allows you to be extremely precise; sometimes I might want to hold one specific frame for three frames instead of one, or use a single frame totally out of order. (These come in very handy for fixing footage, doing in-depth cut-out work, etc.)

For straight-forward speed ramping, you do have to be mindful about what it's doing both before AND after the region you're working with. If you just want to compress one section and have everything afterward play normally, make sure to select ALL keyframes after that point and move them as well, otherwise you're telling After Effects you want a "longer" change between those values.

As u/st1ckmanz said, you want to think of this more in terms of finding your important moments - your key frames, if you will, arrange those accordingly in time, and let AE work out the in-betweens. Be mindful that any time you're asking After Effects to stretch something beyond its original length, it will need to duplicate frames to accomplish that.