Hola! It has the meaning that the person that says that is in pain? Because if I try from English a literal translation in greek it doesn't make any sense..
This is actually a weird sentence without context.
What is understood is that the "la" represents something like "tristeza" "angustia", a fem.noun in Spanish.
And what the sentence means is that
"Everyday, my heart swallows it (the sorrow)"