Its converting an encoded steam to HDMI output. There aren't enough pins on a lightning connector to directly output HDMI and even if it could, you still need a transceiver somewhere (not particularly trivial in terms of space or power to stuff in the phone).
The SoC is most likely a little ARM core to manage stuff (Cortex-M0/3) with a video decoder and HDMI transceiver, the 256MB of ram is there primarily for the decoder to use. It also needs to mux the audio stream into the HDMI encoding.
As for 'it runs iOS', your really splitting hairs, its pretty normal for the master devise to load a slave device with its firmware when plugged in (rather than storing the firmware in flash on the slave). The fact that they would use some knocked down version of iOS isn't terribly surprising. Embedded versions of more powerful OSs are used all the time; There is a non-zero chance your stove and microwave are 'running linux'.