This app was mentioned in 6 comments, with an average of 1.50 upvotes
Usually there is a sound delay on anything like Airplay. You could record right onto your phone if you have android using:
These two apps have options to change video resolutions, bit rate, where you're recording to, or what audio device you want to use (Mic, Internal [ROOT ONLY], both, or none.). If you want to be able to hear your audio while playing, you will need to use the Mic audio option. If you pick any other option you will not be able to hear any audio, but it will be picked up in the recording.
I'm not sure if you absolutely need root for Shou.tv, but rooting for Screen Stream Mirroring is absolutely needed.
Shou.tv is a similar app for those interested in these types of things. Requires root, however.
EDIT: Also, Shou.tv also works on 4.4.x, while MirraTV only can stream from 5+.
You can record it with Shou.
You can record it with Shou.