To 'Walk Spanish' is an American idiom first recorded in 1895, meaning 'to give the bum's rush'. It originally referred specifically to forcing someone to walk by grabbing the collar of their shirt and the back of their trousers and lifting so that they are forced forwards on tiptoe. This practice was said to be common among the pirates of the Spanish Main, when helping reluctant prisoners to walk the plank. Early accounts record that this was accompanied by blows to the behind, making it even less dignified than the modern-day version.
There is also the distinct possibility that this is a false etymology, and that this is nothing more than a derogatory term; in the not-too-distant past it was common to use racial slurs in idioms, from Spanish castle to Dutch courage to French leave.
It has since come to be used for any situation in which a person is moved along against their will -- especially if physical force is used, but you could also refer to someone who is leaving their office after being fired as walking Spanish, or a politician leaving office.
This idiom has since largely fallen out of popular usage, but can be found in older songs and writings. Perhaps the best known recent use of the phrase is Tom Waits' 1985 song Walking Spanish off of his Rain Dogs album; in this case walking Spanish refers to a man walking down death row.