I know I'm missing something fundamental here, but you can imagine the Google results I get trying to find this out!
Shouldn't my phone cameras recorded fps/exposure be way way waayy much slower than 38000 flashes every second?
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Figure 1. Image source: Adafruit.
The data is modulated on the 38 kHz carrier as shown in the image above. The 38 kHz is transmitted in bursts and it's the bursts your camera is detecting. You are correct that your camera's sensor will integrate many 38 kHz pulses in one video exposure "frame".
Having the carrier frequency makes the system much more robust as the receiver can be set up to look for rapid changes in illumination levels. This is a big help in making the system work even with high background radiation such as sunlight.
The linked article is worth a read.
Remotes don't transmit continuously. The output signal looks like bits modulating a 38kHz signal, then a pause like a few tenths of a second, then another transmission. So it'll look like it blinks, but that's probably not an effect of the 38kHz modulation, rather the pauses between retransmissions.