Haskell function incompatible type and definition

Would it be easier if escapeChar were a top-level definition using pattern matching:

escape :: String -> String
escape = concatMap escapeChar

escapeChar :: Char -> String
escapeChar '<' = "&lt;"
escapeChar '>' = "&gt;"
escapeChar ch  = [ch]

[ch] is a singleton list, it turns ch :: Char into a [ch] :: String.

In Haskell you can remove/add an argument from/to each side (eta conversion). escape is the eta reduced form of

escape :: String -> String
escape str = concatMap escapeChar str

Just like, if you wanted to define a synonym for (+) you have equivalent ways of writing it. I feel like the add = (+) is clearest, you are identifying the two functions. The arguments are the same on both sides so we don't specify them.

add :: Int -> Int -> Int
add     = (+)
add a   = (+) a
add a   = (a +)
add a b = (+) a b
add a b = a + b

These are equivalent ways of writing escape:

escape = concat . map escapeChar

escape str = concat (map escapeChar str)

According to the type, escape is a function that takes a String. But it seems that in in the fuction definition it does not receive any argument. How does this work?

concat . map escape returns a function. That function will take a string and process it.

What is the relationship between escapeChar and c? How is that relationship established? Does c coming right after escapeChar have a meaning?

Yes, it is the first (and only) parameter of the function. It is a Character, and the escapeChar function maps that Char on a String. The let clause thus defines a function escapeChar :: Char -> String that will then be used in concat . map escape (or perhaps better concatMap escape). This will map each Char of the given String to a substring, and these are then concatenated together as a result.

the map function has a signature (a -> b) -> ([a] -> [b]) this means that the map function takes in a function(the escapeChar function) and returns a function that converts a list using that function(the escapeChar function). map escapeChar returns a function that converts a string using the escapeChar function on each character in the string.