I found the following to be working.
From what I can gather, one problem was, that the required encoding was form, not json. Another problem was, that from the request you posted, you did not send any body, meaning information on what you want to get back from the API. As it is a pain to type all that out, I typically copy the curl command from chrome and paste it here to get the body formatted nicely.
If you plan on doing this frequently, I recommend putting your email adress or something in the headers of your request, so that Sysadmins have a chance of contacting you if this behavior is not wanted.
require(httr)
headers = c(
`Email` = 'johndoe@company.com'
)
data = list(
`Culture` = 'nl-NL',
`Skip` = '0',
`MaxTotal` = '18',
`ProjectGroup` = '',
`Filters[0][FilterFieldId]` = '6',
`Filters[0][FilterType]` = 'Textbox',
`Filters[1][FilterFieldId]` = '10',
`Filters[1][FilterType]` = 'Checkbox',
`Filters[2][FilterFieldId]` = '7',
`Filters[2][FilterType]` = 'Radiobutton',
`Filters[3][FilterFieldId]` = '5',
`Filters[3][FilterType]` = 'MinMaxSliderInt',
`Filters[3][Values][0][Id]` = '1',
`Filters[3][Values][0][Value]` = '19200',
`Filters[3][Values][1][Id]` = '2',
`Filters[3][Values][1][Value]` = '2600000',
`Filters[4][FilterFieldId]` = '1',
`Filters[4][FilterType]` = 'MinMaxSliderDouble',
`Filters[4][Values][0][Id]` = '1',
`Filters[4][Values][0][Value]` = '1.2',
`Filters[4][Values][1][Id]` = '2',
`Filters[4][Values][1][Value]` = '9',
`Filters[5][FilterFieldId]` = '2',
`Filters[5][FilterType]` = 'Checkbox',
`Filters[6][FilterFieldId]` = '3',
`Filters[6][FilterType]` = 'MinMaxSliderInt',
`Filters[6][Values][0][Id]` = '1',
`Filters[6][Values][0][Value]` = '5',
`Filters[6][Values][1][Id]` = '2',
`Filters[6][Values][1][Value]` = '120',
`Filters[7][FilterFieldId]` = '4',
`Filters[7][FilterType]` = 'Checkbox',
`Filters[8][FilterFieldId]` = '8',
`Filters[8][FilterType]` = 'Checkbox',
`PageId` = '1219',
`OverviewTypeId` = '3',
`WhiteLabelCode` = '',
`OrderBys[0][Id]` = '5',
`OrderBys[0][Value]` = 'Openstaand bedrag',
`OrderByDescending` = 'false'
)
res <- httr::POST(url = 'https://www.geldvoorelkaar.nl/umbraco/api/ProjectApi/GetProjects',
httr::add_headers(.headers=headers),
body = data,
encode = "form")
result <- content(res)