Neither do people put

"Neither does anyone pour" or "People never put"

new wine

This refers to wine that has not fermented yet. Alternate translation: "unfermented wine" or "grape juice" or "unfermented fruit juice"

This refers to wineskins that have stretched and dried out because they were already used for fermenting wine.

"wine bags" or "skin bags." These were bags made out of animal skins.

"and this will ruin the wineskins and spill the wine"

When the new wine ferments and expands, old wineskins tear open because they can no longer stretch out.

"new wineskins" or "new wine bags." This refers to wineskins that no one has used.

"this will keep safe both the wineskins and the wine"