What does "This counts as evolving that Pokémon" mean, in game terms?
It counts towards the one evolution per turn for that Pokémon (with exceptions for Trainers and Pokémon Powers that break that rule) and removes Special Conditions on that Pokémon.
Source: PUI Rules Team (2003-09-26)

Relevant cards

Note: All applicable cards may not be listed below.
BW10_105
BW10_85
BW5_100
DP4_102
EX13_90
EX2_88
EX9_83
HGSS2_82
POP5_7
POP8_10
SM1_129
SM2_165
SM7_142
SWSH1_180
SWSH1_180
XY5_135