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, which are then claimed by "Rebooting", which first becomes available after buying a single Singularity. Afterward, 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.
The formula for MetaBits gained after a reboot is: (Entropy + Ideas)^(1/3) / 10000 - 1 (More accurately, replace the 1/3 with 0.3333333333333333 and you'll have the exact equation, which this equation is confirmed by the source code that SirH searched through)