What were the children of Yisra'el commanded to bring to the door of the Tent of Meeting in Leviticus 17:5 and why?
Answer From The Annual Reading Schedule - Week 27 Notes For Leviticus: Their zevach (in the Hebrew). This Hebrew word, zevach, means "sacrifice" or "thing of sacrifice" or something similar. The ISR Scriptures 1998 version and possibly some other versions derived from the Scriptures version may incorrectly translate zevach as merely "slaughterings". However, a "slaughtering" does not carry the same connotations in the English as "sacrifice" does. A slaughter is merely a killing of an animal. However, a sacrifice is a specific killing of an animal which carries with it specific laws and regulations. The "laws of the zevach" are discussed in more detail in the reading notes for Week 16 of the Annual Reading Schedule, specifically the question "Should believers today take a lamb on Passover and perform the Passover with all its' laws and ordinances, including the blood on the doorposts, etc., or not?". Any killing of any animal might be called a "slaughtering" whereas a "sacrifice" is a specific slaughtering in which those "laws of the zevach" apply.
The reason why they had to bring their sacrifices to the door of the Tent of Meeting is shown in Leviticus 17:7 which states that it was to keep them from offering sacrifices to demons. It seems to me that whatever they were doing out in the open fields was more than just killing an animal, but actually doing sacrifices to demons in some cases. The reason why we know that Leviticus 17:5 is not instituting a new requirement for every single animal ever slaughtered to be brought to the priests is because then that would mean nobody could ever eat meat during the times of the 1st and 2nd Temple unless they were in Yerushalayim where the temple and priests were. However, we know this is not the case.
It would also mean nobody could eat meat today because we have no Temple functioning today with sons of Aharon (Aaron). We know this also is not the case and that meat is permitted to be eaten, so that rules out the idea that this verse is prohibiting any and all slaughterings of animals.
The word in Leviticus 17:5 is not just "slaughterings" but sacrifices – zevach. Zevach are specific kinds of slaughterings which have additional laws that apply. See the link a few paragraphs back for details.