Update:
Thanks to @aelwan for catching a bug, and the updated code is below:
library(ggplot2)
library(reshape2)
# read in the data
dfStage = read.csv("reshapeR/Data/stage.csv", header = FALSE, stringsAsFactor = FALSE)
# remove the rows which are min, max, mean & redundant columns
condMMM = stringr::str_trim(dfStage[, 1]) %in% c("Min", "Max", "Mean", "Day")
dfStage = dfStage[!condMMM, 1:13]
dateVars = c("Day", "Jan", "Feb", "Mar", "Apr", "May", "Jun", "Jul", "Aug", "Sep", "Oct", "Nov", "Dec")
colnames(dfStage) = dateVars
# get indices & names of year site combinations
condlSiteYear = grepl("^Daily means", stringr::str_trim(dfStage[, 1]))
condiSiteYear = grep("^Daily means", stringr::str_trim(dfStage[, 1]))
dfSiteYear = dfStage[condlSiteYear,  1, drop = FALSE]
# remove site-year rows from data
dfStage = dfStage[!condlSiteYear, ]
# get the list of sites and years
dfSiteYear$Year = regmatches(dfSiteYear[, 1], regexpr("(?<=Year\\s)([0-9]+)", dfSiteYear[, 1], perl = TRUE))
dfSiteYear$Site = regmatches(dfSiteYear[, 1], 
           regexpr("(?<=(Stage\\s\\(mm\\)\\sat\\s))([A-Za-z\\s0-9\\.]+)", dfSiteYear[, 1], perl = TRUE))
# add the site and years
dfSiteYearLong = dfSiteYear[rep(1:dim(dfSiteYear)[1], each = 31), c("Site", "Year")]
dfStageFinal = cbind(dfStage, dfSiteYearLong)
# reshape
dfStageFinalLong = reshape2::melt(dfStageFinal, id.vars = c("Day", "Site", "Year"), 
                                  measure.vars = dateVars[-1],
                        variable.name = "Month")
dfStageFinalWide = reshape2::dcast(dfStageFinalLong, Day + Month + Year ~ Site, 
                                   value.var = "value")
# cleanup
dfStageFinalWide[, -c(1:3)] = lapply(dfStageFinalWide[, -c(1:3)], as.numeric)
# create a date variable
dfStageFinalWide$Date = with(dfStageFinalWide, 
                             as.Date(paste(Day, Month, Year, sep = "-"), 
                                     format = "%d-%b-%Y"))
# remove the infeasible dates
dfStageFinalWide = dfStageFinalWide[!is.na(dfStageFinalWide$Date), ]
dfStageFinalWide = dfStageFinalWide[order(dfStageFinalWide$Date), ]
# plot the values over time
dfStageFinalLong = 
  reshape2::melt(dfStageFinalWide, id.vars = "Date", measure.vars = unique(dfSiteYear$Site),
       variable.name = "Site")
ggplot(dfStageFinalLong, aes(x = Date, y = value, color = Site))+
  geom_line() + theme_bw() + facet_wrap(~ Site, scale = "free_y") 
This leads to the picture below:

Original answer:
This example requires a fair amount of data munging skills. You basically have to note the repeating patters in the data -- the data are site-year measurements organized as day x month tables. 
Recipe:
Here is a recipe for creating the desired dataset:
1. Remove the rows & columns in the data that are redundant.
2. Extract the rows that identify the year and the site of the table using pattern matching (grep).
3. From the longer string, extract the year and site name using regular expressions (regexpr and regmatches).
4. Find the starting row indices of the tables for each site-year combination and assign the site-year names just extracted to all rows that correspond to that site & year.
5. Now you can go ahead and reshape it into any shape you want. In the code below, the row identifiers are year, month and day, and the columns are the sites.
6. Some cleanup, and you are good to go.
Code:
Here is code for the recipe above:
# read in the data
dfStage = read.csv("reshapeR/Data/stage.csv", header = FALSE, stringsAsFactor = FALSE)
# remove the rows which are min, max, mean & redundant columns
condMMM = stringr::str_trim(dfStage[, 1]) %in% c("Min", "Max", "Mean", "Day")
dfStage = dfStage[!condMMM, 1:13]
dateVars = c("Day", "Jan", "Feb", "Mar", "Apr", "May", "Jun", "Jul", "Aug", "Sep", "Oct", "Nov", "Dec")
colnames(dfStage) = dateVars
# get indices & names of year site combinations
condlSiteYear = grepl("^Daily means", stringr::str_trim(dfStage[, 1]))
condiSiteYear = grep("^Daily means", stringr::str_trim(dfStage[, 1]))
dfSiteYear = dfStage[condlSiteYear,  1, drop = FALSE]
# remove site-year rows from data
dfStage = dfStage[!condlSiteYear, ]
# get the list of sites and years
dfSiteYear$Year = regmatches(dfSiteYear[, 1], regexpr("(?<=Year\\s)([0-9]+)", dfSiteYear[, 1], perl = TRUE))
dfSiteYear$Site = regmatches(dfSiteYear[, 1], 
           regexpr("(?<=(Stage\\s\\(mm\\)\\sat\\s))([A-Za-z\\s0-9\\.]+)", dfSiteYear[, 1], perl = TRUE))
# add the site and years
dfSiteYearLong = dfSiteYear[rep.int(1:dim(dfSiteYear)[1], 31), c("Site", "Year")]
dfStageFinal = cbind(dfStage, dfSiteYearLong)
# reshape
dfStageFinalLong = reshape2::melt(dfStageFinal, id.vars = c("Day", "Site", "Year"), measure.vars = dateVars[-1],
                        variable.name = "Month")
dfStageFinalWide = dcast(dfStageFinalLong, Day + Month + Year ~ Site, value.var = "value")
# cleanup
dfStageFinalWide[, -c(1:3)] = lapply(dfStageFinalWide[, -c(1:3)], as.numeric)
# create a date variable
dfStageFinalWide$Date = with(dfStageFinalWide, 
                             as.Date(paste(Day, Month, Year, sep = "-"), 
                                     format = "%d-%b-%Y"))
# remove the infeasible dates
dfStageFinalWide = dfStageFinalWide[!is.na(dfStageFinalWide$Date), ]
dfStageFinalWide = dfStageFinalWide[order(dfStageFinalWide$Date), ]
# plot the values over time
dfStageFinalLong = 
  melt(dfStageFinalWide, id.vars = "Date", measure.vars = unique(dfSiteYear$Site),
       variable.name = "Site")
ggplot(dfStageFinalLong, aes(x = Date, y = value, color = Site))+
  geom_line() + theme_bw() + facet_wrap(~ Site, scale = "free_y") 
Output:
Here is what the output looks like:
> head(dfStageFinalWide)
  Day Month Year Kumeti at Te Rehunga Makakahi at Hamua Makuri at Tuscan Hills Manawatu at Hopelands Manawatu at Upper Gorge Manawatu at Weber Road Mangahao at Ballance
1   1   Jan 1990                  454                NA                    700                  5133                      NA                     NA                   NA
2   1   Jan 1991                 1002              3643                   1416                    50                    3597                   1836                18160
3   1   Jan 1992                 3490             34239                   8922                  3049                    1221                    417                   NA
4   1   Jan 1993                  404                NA                    396                  3408                      NA                    272                   NA
5   1   Jan 1994                   NA                NA                   3189                   795                      NA                   2321                 1889
6   1   Jan 1995                16548              1923                  69862                  4808                      NA                   6169                   94
  Mangapapa at Troup Rd Mangatainoka at Larsons Road Mangatainoka at Pahiatua Town Bridge Mangatainoka at Tararua Park Mangatoro at Mangahei Road Oruakeretaki at S.H.2 Napier
1                  9406                         2767                                   NA                           NA                       6838                         2831
2                  4985                         2479                                  823                         1078                         76                          105
3                   478                         3665                                 1415                          210                        394                         8247
4                  6394                         1298                                   NA                         2668                       3837                         1878
5                 14051                         3561                                   NA                         2645                        807                           NA
6                    NA                         1057                                 7029                         4497                         NA                           NA
  Raparapawai at Jackson Rd Tamaki at Stephensons Tiraumea at Ngaturi
1                      5189                 50444               17951
2                       345                   416                3025
3                      1364                  5713                1710
4                      3457                 28078                8670
5                       199                    NA                 292
6                        NA                    NA               22774
And a picture to bring it all together.
