You sacrifice all potential of one Pokemon to boost the potential of another (max of 3 upgrades per card). The tributed card effectively because useless and may as well be sold while the upgraded card gains more stats than another copy of it would at fewer or no hearts.
So say I tribute three different copies of my numerous Magikarp collection to upgrade my Togekiss over the course of finding 3 tribute opportunities. My level 10 Togekiss with 3 hearts currently has 23 HP, 15 ATK, and 7 DEF while my other level 10 Togekiss with 0 hearts has 16 HP, 11 ATK, and 5 DEF.
You may not see a difference in level 1 pokemon but it increases the stats it gains as it grows, similar to being a later stage evolution in an evolutionary line (e.g. Charizard vs Charmander).