"from the people": and not from others (i.e., gentiles), and not from his fellow Cohanim.
"from the slaughterers of the slaughtering": to exclude (from priestly gifts) a treifah (an organically defective animal).
"from the slaughterers of the slaughtering": The time of the slaughtering is the criterion. From here they ruled: If a stranger converted and he had a cow — If it were slaughtered before he converted, he is exempt (from priestly gifts); if after he converted, he is liable. If the matter is in doubt (safek), he is exempt, and "the burden of the proof is upon him who would remove (the object) from his neighbor's domain.