How does SegWit reduce transaction size, when the signature is simply moved to another part of the transaction? This video explains that the signature is moved out of the input section and moved to the new one called the witness. From what I can tell, the witness signature is still contained in this transaction - how would a simple reordering of information magically create more space?

I've perused other articles and videos and none of them seem to address this "magic".
TL;DR:
SegWit does not reduce the transaction size, if you're referring to the raw byte length of transactions. Instead it introduces block weight as a new metric that does not directly correspond to the raw byte length of transactions, but treats witness data as having less weight than other parts of the transaction.



The limit for Bitcoin blocks has been changed with the activation of segwit. Blocks used to be limited to 1,000,000 bytes (1MB). Since segwit they are limited to 4,000,000 weight units.

In calculating the weight of a transaction, bytes are weighed differenty depending on whether they are part of the witness or not:


A non-witness byte weighs four weight units.
A witness byte weighs one weight unit.


This has the effect that a non-segwit transaction contributes exactly the same portion of the limit as before. E.g. the raw bytelength of a P2PKH transaction with one input und two outputs is 222 bytes, and it therefore weighs 888 WU, i.e. 222B / 1,000,000 B = 888 WU / 4,000,000 WU. This means that for non-segwit transactions, the block weight limit has exactly the same effect as the blocksize limit had before, and it is backwards compatible.

However, for segwit transactions the weight is not a quadruple of the raw transaction size. E.g. the raw bytelength of a P2SH-P2WSH 2-of-3 multisig transaction with one input and two outputs is 409 bytes, but its weight 868 WU as a large portion of the transaction input is witness data. A segwit transaction will therefore take a smaller portion of the weight limit than its raw bytelength would suggest.

To allow easier comparison to legacy fee rates, the block weight is often expressed as "virtual size" in "virtual bytes" or "vbytes". The virtual size is calculated by dividing the weight of a transaction by four and rounding up to the full integer. For non-segwit transactions, the raw bytelength and virtual size are equal.

In conclusion, the raw byte length of blocks can now exceed 1,000,000B, but the virtual size cannot exceed 1,000,000vB.