How do these speed up the transaction in any way? I thought every transaction gets verified by a miner and the higher fee the higher priority.
The first accelerator is operated by a mining pool... they reserve 100 slots every hour for "free"... as long as your transaction does not have unconfirmed inputs, and a fee of at least 10 sats/byte... you can try and nab one of those 100 slots. If successful, they'll include your transaction regardless of your fee (as long as it is 10+ sats/byte of course!)
The 2nd one supposedly has links with mining pools and can get your transaction included for free (if under 500 bytes, usually a max of like 3-4 inputs)... or $4 if above 500 bytes in size...