It is because eating non-movable things has been allowed by Prajapati (Brahma or the lord of the beings).
As per the scripture:
What is sthavara (immovable i.e. vegetables, plants, etc.) is the
food of jangama (movable i.e. animals) [Manu. Smrt. - 5.29]
Why it is so? Because, if we don't eat something then we will die. Scriptures allow eating anything whether it's movable(animals) or immovable (plants) when life is in danger:
One can eat meat when his life is in danger. (i.e. if the condition is
such that if you don't eat meat you will die.) [Manu. Smrt. - 5.27]
For the sustenance of life Prajapati created everything. So both
movable and immovable are the food of prana (life force) [Manu.
Smrt. - 5.28]
Having stated that, the scripture has heavily instructed in the favor of not eating meat as it includes causing pain to animals please see here. Now, because we can't eat meat, only plant food is left. So to sustain our lives we have to eat plant food anyway.
Now the question is, don't plants feel pain when they are uprooted or their fruits, leaves, vegetables are plucked? Well, may be they do. But in case of plants, the pain, suffering and self identification is certainly less than the animals. So it involves less karma.
Why can't I just offer animal flesh to Krishna and then eat it?
Your priest is partially right. You can certainly offer meat to gods and eat it (Manu 5.27), but this doesn't mean offering to Krishna. This means offering it to certain higher beings like manes and demigods (in the mode of tamas). In that case proper rite and ritual must be done with proper mantras. Simple mental offering wouldn't work.
Krishna being God Himself doesn't need any kind of food or offering for His sustenance. Whatever a devotee offers Him as a token of love and devotion He accepts it. But other demigods and manes being another form of existence need certain things for their satisfaction which can be offered to them through yajna or sacrifice. Eating meat by offering to them in that way has been accepted:
Irrespective of whether one has bought the meat, killed the animal
himself or has received from another person, there is no fault in
eating meat when it has been offered to the gods and the manes. [Manu -
But anyway, offering food to God before eating it is certainly a good habit because food may contain many other faults apart from it being veg or non-veg.
Edit: the answer has been updated as the question has changed