To excite or enthuse.
To 'ginger up' and 'to get someone's ginger up' appear to be essentially the same phrase. That second form appears in print the earliest, in Thomas Haliburton's Attaché, or Sam Slick in England, 1843:
"Curb him [a horse], talk Yankee to him, and get his ginger up."
This clearly harks back to a well-known, if rather unpleasant practice, recorded here by the appropriately named Francis Grose, in A Classical Dictionary of the Vulgar Tongue, 1785:
"To feague a horse, to put ginger up a horse's fundament, to make him lively and carry his tail well."
On the face of it, the link between the practice and 'gingering up' seems clear. Perhaps that's jumping the gun, as other evidence doesn't support that link. Despite Haliburton's use of 'get his ginger up' in 1843, that phrase, and simply 'ginger up', disappear from use until the late 19th century. When it returns to the language, it is in citations from the USA relating to baseball. The earliest of these is from the Evening News, Lincoln, Nebraska, Aug 1895:
"It is really too bad that just when we had our appetites all fixed for the enjoyment of a baseball game Jupiter Pluvius had to ginger up a little and get in the game himself."
(Note: It rained - in Roman mythology Jupiter Pluvius was the reliever of droughts)
There are many citations following 1895, all relating to baseball. It may be that someone revived the equine reference, but it seems just as likely that the phrase was coined independently of that. 'Gin up', meaning enliven, was in use in the USA before 1895.
The related term 'ginger group', comes later. That has the same root meaning, i.e. a group of people who aim the 'ginger up' others. The best-known early use of this is the UK 'Liberal War Committee', established in January 1916, the 'Ginger Group' of Liberals who helped Lloyd-George to become Prime Minister.