Spell out the Revu'a

Dyalog APL, 2 bytes

,\

Cumulative reduce by concatenate. Try it here.

The formatting of the output is nicer when you prefix a ⍕, but it clearly shows the correct order without.

JavaScript (ES6), 27 26 25 bytes

Saved one byte thanks to @nicael and @MartinBüttner, one thanks to @Neil

x=>x.replace(/.?/g,"$` ")

Takes advantage of some built-in features of JS's .replace function. Specifically, in the replacement, $` becomes everything preceding the matched character. Using the regex /.?/g rather than /./g means it also matches the empty string at the end.

Japt, 10 4 bytes

I didn't realize that a cumulative reduce would be so useful in this case. :-)

UŒ+

Outputs as an array, comma-separated by default. If this is not allowed, use this 6-byte code instead:

U¬å+ ·

Try it online!

How it works

      // Implicit: U = input string
U¬    // Split U into chars.
  å+  // Cumulative reduce: loop through each item in the array, concatenating it to the total.
      // ["t","e","s","t"] => ["t","te","tes","test"].
      // Implicit: output last expression