What is the proper ICAO phraseology to activate a flight plan?
The UK CAA Radiotelephony Manual (CAP 413, Edition 22, May 2016) has an example of how to activate a flight plan in chapter 3, section 3.36 (page 92 in the PDF).


  Aircraft G-CD, departed Seton at 38 request activate flight plan
  ATC G-CD, departure time 38 will activate flight plan


There is no indication whether this is for VFR or IFR flight plans, so it should apply to both if a filed flight plan wasn't activated at departure.

I also pulled out my (fairly recent) radiotelephony textbook, which covers EASA up to and including PPL including in English (and thus should be pretty close to the official ICAO phraseology), to double-check what it says. (I don't often fly with an actual flight plan active.)


  Aircraft ATC callsign, aircraft callsign
  ATC aircraft callsign, ATC callsign, go ahead*
  Aircraft aircraft callsign, activating VFR flight plan, departed airport, hour hour minute minute, to destination airport
  ATC aircraft callsign, activating VFR flight plan, hour hour minute minute
  Aircraft aircraft callsign


For example, Smallprop 123 flying VFR in airspace controlled by Somewhere Control, with a previously filed but not yet activated flight plan:


  Aircraft Somewhere Control, Smallprop 123
  ATC Smallprop 123, Somewhere Control, go ahead*
  Aircraft Smallprop 123, activating VFR flight plan, departed Hometown, one two tree five, to destination Springfield
  ATC Smallprop 123, activating VFR flight plan, one two tree five
  Aircraft Smallprop 123


*) the "go ahead" may be omitted

The exception to this is if you are departing from a controlled airport, in which case a flight plan filed prior to departure is activated automatically. Activating a flight plan manually is only required when departing from uncontrolled airports.

To close a flight plan, the phraseology after establishing contact with ATC is:


  Aircraft Smallprop 123 position Wintermeadow closing VFR flight plan from Hometown to Springfield
  ATC Smallprop 123, flight plan closed at time one tree tree zero
  Aircraft Flight plan closed at time one tree tree zero, Smallprop 123