MetaBits are the prestige currency of the game and are used to improve the Reality Engine the simulation runs on. You gain unclaimed MetaBits by gaining more Ideas and Entropy. They are claimed by "Rebooting", which first becomes available after buying a single Singularity. Afterwards, a Reboot can be initiated at any time.
The total amount of MetaBits earned is the "simulation level," which passively increases the production multiplier by itself. This is independent of the amount of MetaBits on hand, which can be spent in the Reality Engine tree to increase production, click value, or offline production value.
It appears that the formula for MetaBits gained after a reboot is (Total entropy + ideas earned/ 1 Trillion)^(1/3) - MetaBits gained from previous reboots. This means that in order to double your all-time MetaBits, you need to multiply total entropy + ideas by 8.