The "connects to a standard faucet" connection..........Does it attach where the "aerator" is, on the "standard faucet"?
If so, unscrew the aerator from the pictured faucet, and check out the differences.
If you need a new adapter, go to a plumbing supply house, and take all your parts with you so they can help you adapt it.
Not the plumbing section @ the Homeless Depot, or Lowes, unless you are comfortabl with picking one out by eyesight from all the "sealed packages"!