Smartened up, in a showy or garish way.
If you were to tell anyone in the UK that they were 'gussied up' you would probably get back a blank stare - the term is little known there. In the USA you might get a less welcome response, as it is at best a back-handed compliment, referring as it does to a somewhat lurid 'trying too hard' appearance.
It would be nice at this juncture to be able to point to some grande dame famous for her kitsch apparel who was the source of the expression. Sadly, the plot is thicker than that and this phrase's origin is uncertain. There are some clues though, so I'll itemize the evidence and leave you to make what you can of it.
First, the word 'gussie'. This was first used in Australia in the early 20th century as a name for a foppish dandy. The Australian feminist novelist Miles Franklin used it in her best-known work My Brilliant Career, 1901, to describe a soppy Lothario to whom she gives the unambiguous but mocking name of Everard Grey:
"I'll show him [Everard] I think no more of him than of the caterpillars on the old tree there. I'm not a booby that will fall in love with every gussie I see. I hate and detest men!"
'Gussie' was a contraction of the name Augustus, which was the generic name for Roman emperors. In some contexts the name might conjure up thoughts of imperial grandeur, but not so here. It wouldn't have been a name commonly given to horny-handed Australian cobbers and 'gussie' was clearly meant to denote effeminacy, just as 'jessie' does now in Australia and the UK. The New Zealand lexicographer Sidney Baker helpfully removes the need for speculation by defining the word in the Popular Dictionary of Australian Slang, 1941:
Gussie, an effeminate or affected man.
'Gussie' is also found in US publications, from a slightly later date, with the same meaning.
Onward, to 'gussie up'. This appears to be of American origin and, as I've mentioned, has largely stayed in the US. The first example that I can find of it in print is in the July 1945 edition of The Rotarian magazine:
The sky is a giddy blue, gussied up with flounces of billowy white clouds.
How, why, or even if 'gussied up' evolved from 'gussie' we don't now know. It could be as simple as the expression just derived as meaning 'dressed like an effeminate man'.
Also playing a part in this is the US tennis player Gertrude Augusta "Gussie" Moran. Known as Gorgeous Gussie, she was a top player in the late 1940s but, after appearing at Wimbledon and other tournaments in frilly knickers, she is now better remembered for her underwear than her overarm. She wasn't the source of the phrase but if anyone could be said to have 'gussied up' the previously staid tennis uniform it was she and the correspondence of her name and appearance without doubt brought the expression into wider use.